Polynomials afgFrom Wikipedia, the free encyclopedia

Contents

1 Abel polynomials 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 AbelRuni theorem 32.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Lower-degree polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Quintics and higher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Actuarial polynomials 73.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Additive polynomial 84.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 The ring of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 The fundamental theorem of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Alexander polynomial 105.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Computing the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Basic properties of the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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5.4 Geometric signicance of the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Relations to satellite operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.6 AlexanderConway polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 Relation to Khovanov homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Algebraic equation 146.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Algebraic variety 177.1 Introduction and denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.1.1 Ane varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.2 Projective varieties and quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . . 197.1.3 Abstract varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.1 Subvariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2.2 Ane variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2.3 Projective variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Isomorphism of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.5 Discussion and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.6 Algebraic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 All one polynomial 258.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Almost linear hash function 279.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Alternating polynomial 2910.1 Relation to symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Vandermonde polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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10.2.1 Ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Unstable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Angelescu polynomials 3211.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12 Appell sequence 3312.1 Equivalent characterizations of Appell sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.2 Recursion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.3 Subgroup of the Sheer polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.4 Dierent convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13 Bell polynomials 3713.1 Complete Bell polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.2 Combinatorial meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.3.1 Stirling numbers and Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3.2 Touchard polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3.3 Convolution identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.4 Applications of Bell polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.1 Fa di Brunos formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.2 Moments and cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.3 Representation of polynomial sequences of binomial type . . . . . . . . . . . . . . . . . . 40

13.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

14 Bernoulli polynomials 4314.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.1.1 Explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.2 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.3 Representation by a dierential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.4 Representation by an integral operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.2 Another explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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14.3 Sums of pth powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.4 The Bernoulli and Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.5 Explicit expressions for low degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.6 Maximum and minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.7 Dierences and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14.7.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.7.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

14.8 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.9 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.10Relation to falling factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.11Multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.12Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.13Periodic Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15 Bernstein polynomial 5315.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.4 Approximating continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

15.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

16 BernsteinSato polynomial 5816.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

17 Binomial 6117.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Operations on simple binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

18 BoasBuck polynomials 6318.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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19 BollobsRiordan polynomial 6419.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

20 Bombieri norm 6620.1 Bombieri scalar product for homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . 6620.2 Bombieri inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.3 Invariance by isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.4 Other inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

21 Boole polynomials 6921.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

22 Bracket polynomial 7022.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

23 Bring radical 7123.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.1.1 Principal quintic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.2 BringJerrard normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.1.3 Brioschi normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

23.2 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.3 Solution of the general quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.4 Other characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

23.4.1 The HermiteKroneckerBrioschi characterization . . . . . . . . . . . . . . . . . . . . . . 7423.4.2 Glassers derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.4.3 The method of dierential resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.4.4 DoyleMcMullen iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

23.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

24 Bzout matrix 8224.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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24.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

25 Caloric polynomial 8525.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

26 Casus irreducibilis 8626.1 Formal statement and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.2 Solution in non-real radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.3 Non-algebraic solution in terms of real quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.4 Relation to angle trisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.5 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27 Cavalieris quadrature formula 8927.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

27.1.1 Negative n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.1.2 n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.1.3 Alternative forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

27.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.4.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

27.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28 Characteristic polynomial 9628.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5 Characteristic polynomial of a product of two matrices . . . . . . . . . . . . . . . . . . . . . . . . 9828.6 Secular function and secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

28.6.1 Secular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.6.2 Secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

28.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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29 Coecient 10029.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.2 Examples of physical coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30 Coecient diagram method 10230.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

31 Complex conjugate root theorem 10431.1 Examples and consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

31.1.1 Corollary on odd-degree polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2 Simple proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

32 Complex quadratic polynomial 10732.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.2 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

32.2.1 Between forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.2.2 With doubling map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

32.3 Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.4 Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.6 Critical items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

32.6.1 Critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.2 Critical value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.3 Critical orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.4 Critical sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.6.5 Critical polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.6.6 Critical curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

32.7 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11132.7.1 Parameter plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.7.2 Dynamical plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

32.8 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8.1 Derivative with respect to c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8.2 Derivative with respect to z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11532.8.3 Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

32.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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33 Constant function 11833.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12033.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

34 Constant term 12134.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

35 Content (algebra) 12335.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12335.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

36 Continuant (mathematics) 12436.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12436.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12436.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

37 Cubic function 12637.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.2 Critical points of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13037.3 Roots of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

37.3.1 The nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13037.3.2 General formula for roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13237.3.3 Reduction to a depressed cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13337.3.4 Cardanos method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13337.3.5 Vietas substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13537.3.6 Lagranges method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.3.7 Trigonometric (and hyperbolic) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.3.8 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.3.9 Geometric interpretation of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

37.4 Collinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38 Cyclotomic polynomial 14738.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

38.2.1 Fundamental tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14938.2.2 Easy cases for the computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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38.2.3 Integers appearing as coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15038.2.4 Gausss formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15038.2.5 Lucass formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

38.3 Cyclotomic polynomials over Zp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.4 Prime Cyclotomic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

39 Degree of a polynomial 15439.1 Names of polynomials by degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.3 Behavior under polynomial operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

39.3.1 Behaviour under addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.3.2 Behaviour under scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.3.3 Behaviour under multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.3.4 Behaviour under composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

39.4 Degree of the zero polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.5 Computed from the function values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.6 Extension to polynomials with two or more variables . . . . . . . . . . . . . . . . . . . . . . . . . 15739.7 Degree function in abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

40 Delta operator 15940.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.2 Basic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

41 Denisyuk polynomials 16141.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

42 Derivation of the Routh array 16242.1 The Cauchy index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.2 The Routh criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16442.3 Sturms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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43 Descartes rule of signs 16843.1 Descartes rule of signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

43.1.1 Positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.2 Negative roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.3 Example: real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.4 Complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.1.5 Example: zero coecients, complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . 169

43.2 Special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

44 Dickson polynomial 17144.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17144.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.3 Links to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.4 Permutation polynomials and Dickson polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

45 Dierence polynomials 17445.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17445.2 Moving dierences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17445.3 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17545.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17545.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

46 Discriminant 17646.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17646.2 Formulas for low degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.3 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17846.4 Quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18046.5 Discriminant of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18046.6 Nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

46.6.1 Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18246.6.2 Cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18246.6.3 Higher degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

46.7 Discriminant of a polynomial over a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . 18246.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

46.8.1 Discriminant of a conic section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.8.2 Discriminant of a quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.8.3 Discriminant of an algebraic number eld . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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46.9 Alternating polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18446.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18446.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

47 Divided power structure 18647.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.3 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

48 Division polynomials 18848.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18848.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

49 Ehrhart polynomial 19149.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19149.2 Examples of Ehrhart Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19149.3 Ehrhart Quasi-Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19249.4 Examples of Ehrhart Quasi-Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.5 Interpretation of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.6 Ehrhart Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

49.6.1 Ehrhart series for rational polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.7 Toric Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

50 Eisensteins criterion 19650.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

50.1.1 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.3 Basic proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.4 Advanced explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.5 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

50.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20050.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20050.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

51 Equally spaced polynomial 201

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51.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20151.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

52 Equioscillation theorem 20252.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

53 Exponential polynomial 20353.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

53.1.1 In elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20353.1.2 In abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

53.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20453.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20453.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

54 External ray 20554.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

54.3.1 Dynamical plane = z-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.3.2 Parameter plane = c-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20754.3.3 External angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20954.3.4 Computation of external argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

54.4 Transcendental maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20954.5 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

54.5.1 Dynamic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20954.5.2 Parameter rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

54.6 Programs that can draw external rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21154.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21154.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21154.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

55 Faber polynomials 21355.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

56 Factor theorem 21456.1 Factorization of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

56.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

57 Factorization of polynomials 21657.1 Formulation of the question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21657.2 Primitive partcontent factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

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57.3 Square-free factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.4 Classical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

57.4.1 Obtaining linear factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.4.2 Kroneckers method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

57.5 Modern methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21957.5.1 Factoring over nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21957.5.2 Factoring univariate polynomials over the integers . . . . . . . . . . . . . . . . . . . . . . 21957.5.3 Factoring over algebraic extensions (Tragers method) . . . . . . . . . . . . . . . . . . . . 220

57.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22157.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22157.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

58 Factorization of polynomials over nite elds 22258.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

58.1.1 Finite eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22258.1.2 Irreducible polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22258.1.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

58.2 Factoring algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.2.1 Square-free factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.2.2 Distinct-degree factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22458.2.3 Equal-degree factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22558.2.4 Victor Shoups algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

58.3 Rabins test of irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22758.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22858.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22858.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22858.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

59 Fekete polynomial 22959.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23059.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

60 Fibonacci polynomials 23160.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23160.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23260.3 Combinatorial interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23360.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23360.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23460.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

61 Gausss lemma (polynomial) 23561.1 Formal statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23561.2 Proofs of the primitivity statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

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61.2.1 A variation, valid over arbitrary commutative rings . . . . . . . . . . . . . . . . . . . . . 23661.2.2 A proof valid over any GCD domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

61.3 Proof of the irreducibility statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23761.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23761.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

62 GaussLucas theorem 23962.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23962.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23962.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23962.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24062.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24062.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24162.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

63 Generalized Appell polynomials 24263.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24263.2 Explicit representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24263.3 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24363.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24363.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

64 Gould polynomials 24564.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

65 GraceWalshSzeg theorem 24665.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24665.2 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

66 List of polynomial topics 24766.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24766.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

66.2.1 Elementary abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24866.3 Theory of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24866.4 Calculus with polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24966.5 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24966.6 Weierstrass approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24966.7 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24966.8 Named polynomials and polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25066.9 Knot polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25166.10Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25166.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 252

66.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

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66.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25666.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Chapter 1

Abel polynomials

The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

pn(x) = x(x an)n1:The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may beobtained from the Abel sequence in the umbral calculus.

1.1 ExamplesFor a=1, the polynomials are (sequence A137452 in OEIS)

p0(x) = 1;

p1(x) = x;

p2(x) = 2x+ x2;p3(x) = 9x 6x2 + x3;p4(x) = 64x+ 48x2 12x3 + x4;For a=2, the polynomials are

p0(x) = 1;

p1(x) = x;

p2(x) = 4x+ x2;p3(x) = 36x 12x2 + x3;p4(x) = 512x+ 192x2 24x3 + x4;p5(x) = 10000x 4000x2 + 600x3 40x4 + x5;p6(x) = 248832x+ 103680x2 17280x3 + 1440x4 60x5 + x6;

1.2 References Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. (1997). All Polynomials of Binomial Type Are Repre-

sented by Abel Polynomials. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sr. 4 25 (34):731738. MR 1655539. Zbl 1003.05011.

1

2 CHAPTER 1. ABEL POLYNOMIALS

1.3 External links Weisstein, Eric W., Abel Polynomial, MathWorld.

Chapter 2

AbelRuni theorem

Not to be confused with Abels theorem.

x = bpb24ac2a

A general solution to any quadratic equation can be given using the quadratic formula above. Similar formulas existfor polynomial equations of degree 3 and 4. But no such formula is possible for 5th degree polynomials; the realsolution 1.1673... to the 5th degree equation below cannot be written using basic arithmetic operations and nthroots:x5 x+ 1 = 0In algebra, the AbelRuni theorem (also known as Abels impossibility theorem) states that there is no gen-eral algebraic solutionthat is, solution in radicals to polynomial equations of degree ve or higher with arbitrarycoecients.[1] The theorem is named after Paolo Runi, who made an incomplete proof in 1799, and Niels HenrikAbel, who provided a proof in 1823. variste Galois independently proved the theorem in a work that was posthu-mously published in 1846.[2]

2.1 InterpretationThe theorem does not assert that some higher-degree polynomial equations have no solution. In fact, the oppositeis true: every non-constant polynomial equation in one unknown, with real or complex coecients, has at least onecomplex number as a solution (and thus, by polynomial division, as many complex roots as its degree, countingrepeated roots); this is the fundamental theorem of algebra. These solutions can be computed to any desired degreeof accuracy using numerical methods such as the NewtonRaphson method or Laguerre method, and in this way theyare no dierent from solutions to polynomial equations of the second, third, or fourth degrees. The theorem onlyshows that the solutions of some of these equations cannot be expressed via a general expression in radicals.Also, the theorem does not assert that some higher-degree polynomial equations have roots which cannot be expressedin terms of radicals. While this is now known to be true, it is a stronger claim, which was only proven a few yearslater by Galois. The theorem only shows that there is no general solution in terms of radicals which gives the roots toa generic polynomial with arbitrary coecients. It did not by itself rule out the possibility that each polynomial maybe solved in terms of radicals on a case-by-case basis.

2.2 Lower-degree polynomialsThe solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multi-plication, division, and square roots, using the familiar quadratic formula: The roots of the following equation areshown below:

ax2 + bx+ c = 0; a 6= 0

3

4 CHAPTER 2. ABELRUFFINI THEOREM

x =bpb2 4ac

2a:

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, have been known sincethe 16th century.

2.3 Quintics and higherThe AbelRuni theorem says that there are some fth-degree equations whose solution cannot be so expressed.The equation x5 x + 1 = 0 is an example. (See Bring radical.) Some other fth degree equations can be solvedby radicals, for example x5 x4 x + 1 = 0 , which factors into (x 1)(x 1)(x + 1)(x + i)(x i) = 0 .The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannotwas given by variste Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if andonly if its Galois group (over the rational numbers, or more generally over the base eld of admitted constants) is asolvable group.Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations can alwaysbe solved by radicals because the symmetric groups S2, S3 and S4 are solvable groups, whereas Sn is not solvablefor n 5. This is so because for a polynomial of degree n with indeterminate coecients (i.e., given by symbolicparameters), the Galois group is the full symmetric group Sn (this is what is called the general equation of the n-thdegree). This remains true if the coecients are concrete but algebraically independent values over the base eld.

2.4 ProofThe following proof is based on Galois theory (for a short explanation of Arnolds proof that does not rely on priorknowledge in group theory see [3]). Historically, Runi and Abels proofs precede Galois theory and Arnolds. Fora modern presentation of Abels proof see the books of Tignol or Pesic.One of the fundamental theorems of Galois theory states that a polynomial f(x) F[x] is solvable by radicals overF if and only if its splitting eld K over F has a solvable Galois group,[4] so the proof of the AbelRuni theoremcomes down to computing the Galois group of the general polynomial of the fth degree.Let y1 be a real number transcendental over the eld of rational numbersQ , and let y2 be a real number transcendentalover Q(y1) , and so on to y5 which is transcendental over Q(y1; y2; y3; y4) . These numbers are called independenttranscendental elements over Q. Let E = Q(y1; y2; y3; y4; y5) and let

f(x) = (x y1)(x y2)(x y3)(x y4)(x y5) 2 E[x]:Expanding f(x) out yields the elementary symmetric functions of the yn :

s1 = y1 + y2 + y3 + y4 + y5

s2 = y1y2 + y1y3 + y1y4 + y1y5 + y2y3 + y2y4 + y2y5 + y3y4 + y3y5 + y4y5

s3 = y1y2y3 + y1y2y4 + y1y2y5 + y1y3y4 + y1y3y5 + y1y4y5 + y2y3y4 + y2y3y5 + y2y4y5 + y3y4y5

s4 = y1y2y3y4 + y1y2y3y5 + y1y2y4y5 + y1y3y4y5 + y2y3y4y5

s5 = y1y2y3y4y5:

The coecient of xn in f(x) is thus (1)5ns5n . Let F = Q(si) be the eld obtained by adjoining the symmetricfunctions to the rationals (the si are all transcendental, because the yi are independent). Because our independenttranscendentals yn act as indeterminates over Q , every permutation in the symmetric group on 5 letters S5 inducesa distinct automorphism 0 onE that leavesQ xed and permutes the elements yn . Since an arbitrary rearrangementof the roots of the product form still produces the same polynomial, e.g.:

(y y3)(y y1)(y y2)(y y5)(y y4)

2.5. HISTORY 5

is still the same polynomial as

(y y1)(y y2)(y y3)(y y4)(y y5)

the automorphisms 0 also leave f xed, so they are elements of the Galois group G(E/F ) . So we have shownthat S5 G(E/F ) ; however there could possibly be automorphisms there that are not in S5 . However, since therelative automorphism group for the splitting eld of a quintic polynomial has at most 5! elements, it follows thatG(E/F ) is isomorphic to S5 . Generalizing this argument shows that the Galois group of every general polynomialof degree n is isomorphic to Sn .And what of S5 ? The only composition series of S5 is S5 A5 feg (where A5 is the alternating group on veletters, also known as the icosahedral group). However, the quotient groupA5/feg (isomorphic toA5 itself) is not anabelian group, and so S5 is not solvable, so it must be that the general polynomial of the fth degree has no solutionin radicals. Since the rst nontrivial normal subgroup of the symmetric group on n letters is always the alternatinggroup on n letters, and since the alternating groups on n letters for n 5 are always simple and non-abelian, andhence not solvable, it also says that the general polynomials of all degrees higher than the fth also have no solutionin radicals.Note that the above construction of the Galois group for a fth degree polynomial only applies to the general poly-nomial, specic polynomials of the fth degree may have dierent Galois groups with quite dierent properties, e.g.x5 1 has a splitting eld generated by a primitive 5th root of unity, and hence its Galois group is abelian and theequation itself solvable by radicals; moreover the argument does not provide any rational-valued quintic that has S5or A5 as its Galois group. However, since the result is on the general polynomial, it does say that a general quinticformula for the roots of a quintic using only a nite combination of the arithmetic operations and radicals in termsof the coecients is impossible. Q.E.D.

2.5 HistoryAround 1770, Joseph Louis Lagrange began the groundwork that unied the many dierent tricks that had been usedup to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrangeresolvents. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions forequations of fth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusiveproof. The theorem, however, was rst nearly proved by Paolo Runi in 1799, but his proof was mostly ignored.He had several times tried to send it to dierent mathematicians to get it acknowledged, amongst them, Frenchmathematician Augustin-Louis Cauchy, but it was never acknowledged, possibly because the proof was spanning 500pages. The proof also, as was discovered later, contained an error. In modern terms, Runi failed to prove that thesplitting eld is one of the elds in the tower of radicals which corresponds to the hypothesized solution by radicals;this assumption fails, for example, for Cardanos solution of the cubic; it splits not only the original cubic but alsothe two others with the same discriminant. While Cauchy felt that the assumption was minor, most historians believethat the proof was not complete until Abel proved this assumption. The theorem is thus generally credited to NielsHenrik Abel, who published a proof that required just six pages in 1824.[5]

Proving that some quintic (and higher) equations were unsolvable by radicals did not completely settle the matter,because the AbelRuni theorem does not provide necessary and sucient conditions for saying precisely whichquintic (and higher) equations are unsolvable by radicals; for example x5 1 = 0 is solvable. Abel was working on acomplete characterization when he died in 1829.[6] Furthermore, Ian Stewart notes that for all that Abels methodscould prove, every particular quintic equation might be soluble, with a special formula for each equation.[7]

In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvabilityby radicals; Galois paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in termsof the roots of the equation instead of its coecients. Galois then died in 1832 and his paper"Memoire sur lesconditions de resolubilite des equations par radicauxremained unpublished until 1846 when it was published byJoseph Liouville accompanied by some of his own explanations.[6] Prior to this publication, Liouville announcedGalois result to the Academy in a speech he gave on 4 July 1843.[8] According to Allan Clark, Galoiss characterizationdramatically supersedes the work of Abel and Runi.[9]

In 1963, Vladimir Arnold discovered a topological proof of the AbelRuni theorem,[10] which served as a startingpoint for topological Galois theory.[11]

6 CHAPTER 2. ABELRUFFINI THEOREM

2.6 See also Theory of equations Constructible number

2.7 Notes[1] Jacobson (2009), p. 211.[2] Galois, variste (1846). OEuvres mathmatiques d'variste Galois.. Journal des mathmatiques pures et appliques XI:

381444. Retrieved 2009-02-04.[3] Short proof of Abels theorem that 5th degree polynomial equations cannot be solved.[4] Fraleigh (1994, p. 401)[5] du Sautoy, Marcus. January: Impossibilities. Symmetry: A Journey into the Patterns of Nature. ISBN 978-0-06-078941-

1.[6] Jean-Pierre Tignol (2001). Galois Theory of Algebraic Equations. World Scientic. pp. 232233 and 302. ISBN 978-

981-02-4541-2.[7] Stewart, 3rd ed., p. xxiii[8] Stewart, 3rd ed., p. xxiii[9] Allan Clark (1984) [1971]. Elements of Abstract Algebra. Courier Corporation. p. 131. ISBN 978-0-486-14035-3.

[10] Tribute to Vladimir Arnold (PDF).Notices of the AmericanMathematical Society 59 (3): 393. March 2012. doi:10.1090/noti810.[11] Vladimir Igorevich Arnold. 2010.

2.8 References Edgar Dehn. Algebraic Equations: An Introduction to the Theories of Lagrange and Galois. Columbia Univer-

sity Press, 1930. ISBN 0-486-43900-3. Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 John B. Fraleigh. A First Course in Abstract Algebra. Fifth Edition. Addison-Wesley, 1994. ISBN 0-201-

59291-6. Ian Stewart. Galois Theory. Chapman and Hall, 1973. ISBN 0-412-10800-3. Abels Impossibility Theorem at Everything2

2.9 Further reading Peter Pesic (2003). Abels Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. MIT

Press. ISBN 978-0-262-66182-9.

2.10 External links Mmoire sur les quations algbriques, ou l'on dmontre l'impossibilit de la rsolution de l'quation gnrale

du cinquime degr PDF - the rst proof in French (1824) Dmonstration de l'impossibilit de la rsolution algbrique des quations gnrales qui passent le quatrime

degr PDF - the second proof in French (1826) A video presentation on Arnolds proof.

Chapter 3

Actuarial polynomials

In mathematics, the actuarial polynomials a()n(x) are polynomials studied by Toscano (1950) given by the generating function

Xn

a()n (x)

n!tn = exp(t+ x(1 et))

(Roman 1984, 4.3.4), Boas & Buck (1958).

3.1 See also Umbral calculus

3.2 References Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Math-

ematik und ihrer Grenzgebiete. Neue Folge. 19, Berlin, New York: Springer-Verlag, MR 0094466

Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics 111, London: Academic PressInc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 741185 Reprinted by Dover,2005

Toscano, Letterio (1950), Una classe di polinomi della matematica attuariale, Rivista di Matematica dellaUniversit di Parma (in Italian) 1: 459470, MR 0040480, Zbl 0040.03204

7

Chapter 4

Additive polynomial

In mathematics, the additive polynomials are an important topic in classical algebraic number theory.

4.1 DenitionLet k be a eld of characteristic p, with p a prime number. A polynomial P(x) with coecients in k is called anadditive polynomial, or a Frobenius polynomial, if

P (a+ b) = P (a) + P (b)

as polynomials in a and b. It is equivalent to assume that this equality holds for all a and b in some innite eldcontaining k, such as its algebraic closure.Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition thatP(a + b) = P(a) + P(b) for all a and b in the eld. For innite elds the conditions are equivalent, but for nite eldsthey are not, and the weaker condition is the wrong one and does not behave well. For example, over a eld oforder q any multiple P of xq x will satisfy P(a + b) = P(a) + P(b) for all a and b in the eld, but will usually not be(absolutely) additive.

4.2 ExamplesThe polynomial xp is additive. Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem

(a+ b)p =

pXn=0

p

n

anbpn:

Since p is prime, for all n = 1, ..., p1 the binomial coecient (pn) is divisible by p, which implies that

(a+ b)p ap + bp mod p

as polynomials in a and b.Similarly all the polynomials of the form

np (x) = xpn

are additive, where n is a non-negative integer.

8

4.3. THE RING OF ADDITIVE POLYNOMIALS 9

4.3 The ring of additive polynomialsIt is quite easy to prove that any linear combination of polynomials np (x) with coecients in k is also an additivepolynomial. An interesting question is whether there are other additive polynomials except these linear combinations.The answer is that these are the only ones.One can check that if P(x) and M(x) are additive polynomials, then so are P(x) + M(x) and P(M(x)). These implythat the additive polynomials form a ring under polynomial addition and composition. This ring is denoted

kfpg:

This ring is not commutative unless k equals the eld Fp=Z/pZ (see modular arithmetic). Indeed, consider the additivepolynomials ax and xp for a coecient a in k. For them to commute under composition, we must have

(ax)p = axp;

or ap a = 0. This is false for a not a root of this equation, that is, for a outside Fp:

4.4 The fundamental theorem of additive polynomialsLet P(x) be a polynomial with coecients in k, and fw1;:::;wmgk be the set of its roots. Assuming that the roots ofP(x) are distinct (that is, P(x) is separable), then P(x) is additive if and only if the set fw1;:::;wmg forms a group withthe eld addition.

4.5 See also Drinfeld module Additive map

4.6 References David Goss, Basic Structures of Function Field Arithmetic, 1996, Springer, Berlin. ISBN 3-540-61087-1.

4.7 External links Weisstein, Eric W., Additive Polynomial, MathWorld.

Chapter 5

Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coecientsto each knot type. James Waddell Alexander II discovered this, the rst knot polynomial, in 1923. In 1969, JohnConway showed a version of this polynomial, now called the AlexanderConway polynomial, could be computedusing a skein relation, although its signicance was not realized until the discovery of the Jones polynomial in 1984.Soon after Conways reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibitedin Alexanders paper on his polynomial.[1]

5.1 DenitionLet K be a knot in the 3-sphere. Let X be the innite cyclic cover of the knot complement of K. This covering can beobtained by cutting the knot complement along a Seifert surface of K and gluing together innitely many copies ofthe resulting manifold with boundary in a cyclic manner. There is a covering transformation t acting on X. Considerthe rst homology (with integer coecients) of X, denoted H1(X) . The transformation t acts on the homology andso we can consider H1(X) a module over Z[t; t1] . This is called the Alexander invariant or Alexander module.The module is nitely presentable; a presentation matrix for this module is called the Alexander matrix. If thenumber of generators, r, is less than or equal to the number of relations, s, then we consider the ideal generated byall r by r minors of the matrix; this is the zero'th Fitting ideal or Alexander ideal and does not depend on choiceof presentation matrix. If r > s, set the ideal equal to 0. If the Alexander ideal is principal, take a generator; this iscalled an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomialtn , one often xes a particular unique form. Alexanders choice of normalization is to make the polynomial havea positive constant term.Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial alwaysexists, and is clearly a knot invariant, denoted K(t) . The Alexander polynomial for the knot congured by onlyone string is a polynomial of t2 and then it is the same polynomial for the mirror image knot. Namely, it can notdistinguish between the knot and one for its mirror image.

5.2 Computing the polynomialThe following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.Take an oriented diagram of the knot with n crossings; there are n + 2 regions of the knot diagram. To work out theAlexander polynomial, rst one must create an incidence matrix of size (n, n + 2). The n rows correspond to the ncrossings, and the n + 2 columns to the regions. The values for the matrix entries are either 0, 1, 1, t, t.Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, theentry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives theentry, determined by the location of the region at the crossing from the perspective of the incoming undercrossingline.

on the left before undercrossing: t

10

5.3. BASIC PROPERTIES OF THE POLYNOMIAL 11

on the right before undercrossing: 1on the left after undercrossing: ton the right after undercrossing: 1

Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the newn by n matrix. Depending on the columns removed, the answer will dier by multiplication by tn . To resolve thisambiguity, divide out the largest possible power of t and multiply by 1 if necessary, so that the constant term ispositive. This gives the Alexander polynomial.The Alexander polynomial can also be computed from the Seifert matrix.After the work of Alexander R. Fox considered a copresentation of the knot group 1(S3nK) , and introduced non-commutative dierential calculus Fox (1961), which also permits one to compute K(t) . Detailed exposition ofthis approach about higher Alexander polynomials can be found in the book Crowell & Fox (1963).

5.3 Basic properties of the polynomialThe Alexander polynomial is symmetric: K(t1) = K(t) for all knots K.

From the point of view of the denition, this is an expression of the Poincar Duality isomorphismH1X ' HomZ[t;t1](H1X;G) where G is the quotient of the eld of fractions of Z[t; t1] by Z[t; t1], considered as a Z[t; t1] -module, and where H1X is the conjugate Z[t; t1] -module to H1X ie: asan abelian group it is identical to H1X but the covering transformation t acts by t1 .

and it evaluates to a unit on 1: K(1) = 1 .

From the point of view of the denition, this is an expression of the fact that the knot complement is ahomology circle, generated by the covering transformation t . More generally if M is a 3-manifold suchthat rank(H1M) = 1 it has an Alexander polynomial M (t) dened as the order ideal of its innite-cyclic covering space. In this case M (1) is, up to sign, equal to the order of the torsion subgroup ofH1M .

It is known that every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexanderpolynomial of a knot (Kawauchi 1996).

5.4 Geometric signicance of the polynomialSince the Alexander ideal is principal, K(t) = 1 if and only if the commutator subgroup of the knot group is perfect(i.e. equal to its own commutator subgroup).For a topologically slice knot, the Alexander polynomial satises the FoxMilnor condition K(t) = f(t)f(t1)where f(t) is some other integral Laurent polynomial.Twice the knot genus is bounded below by the degree of the Alexander polynomial.Michael Freedman proved that a knot in the 3-sphere is topologically slice; i.e., bounds a locally-at topologicaldisc in the 4-ball, if the Alexander polynomial of the knot is trivial (Freedman and Quinn, 1990).Kauman (1983) describes the rst construction of the Alexander polynomial via state sums derived from physicalmodels. A survey of these topic and other connections with physics are given in Kauman (2001).There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions,there is a way of modifying a smooth 4-manifold by performing a surgery that consists of removing a neighborhood ofa two-dimensional torus and replacing it with a knot complement crossed with S1. The result is a smooth 4-manifoldhomeomorphic to the original, though now the SeibergWitten invariant has been modied by multiplication withthe Alexander polynomial of the knot.[2]

Knots with symmetries are known to have restricted Alexander polynomials. See the symmetry section in (Kawauchi1996). Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.

12 CHAPTER 5. ALEXANDER POLYNOMIAL

If the knot complement bers over the circle, then the Alexander polynomial of the knot is known to be monic (thecoecients of the highest and lowest order terms are equal to 1 ). In fact, if S ! CK ! S1 is a ber bundlewhere CK is the knot complement, let g : S ! S represent the monodromy, then K(t) = Det(tI g) whereg : H1S ! H1S is the induced map on homology.

5.5 Relations to satellite operationsIf a knot K is a satellite knot with companion K 0 i.e.: there exists an embedding f : S1 D2 ! S3 such thatK = f(K 0) where S1 D2 S3 is an unknotted solid torus, then K(t) = f(S1f0g)(ta)K0(t) . Wherea 2 Z is the integer that represents K 0 S1 D2 in H1(S1 D2) = Z .Examples: For a connect-sum K1#K2(t) = K1(t)K2(t) . If K is an untwisted Whitehead double, thenK(t) = 1 .

5.6 AlexanderConway polynomialAlexander proved the Alexander polynomial satises a skein relation. John Conway later rediscovered this in adierent form and showed that the skein relation together with a choice of value on the unknot was enough to deter-mine the polynomial. Conways version is a polynomial in z with integer coecients, denoted r(z) and called theAlexanderConway polynomial (also known as Conway polynomial or ConwayAlexander polynomial).Suppose we are given an oriented link diagram, where L+; L; L0 are link diagrams resulting from crossing andsmoothing changes on a local region of a specied crossing of the diagram, as indicated in the gure.

L+ L- L 0Here are Conways skein relations:

r(O) = 1 (where O is any diagram of the unknot)

r(L+)r(L) = zr(L0)

The relationship to the standard Alexander polynomial is given byL(t2) = rL(tt1) . HereL must be properlynormalized (by multiplication of tn/2 ) to satisfy the skein relation (L+) (L) = (t1/2 t1/2)(L0) .Note that this relation gives a Laurent polynomial in t1/2.See knot theory for an example computing the Conway polynomial of the trefoil.

5.7. RELATION TO KHOVANOV HOMOLOGY 13

5.7 Relation to Khovanov homologyIn Ozsvath & Szabo (2004) and Rasmussen (2003) the Alexander polynomial is presented as Euler characteristic ofa complex, whose homology are isotopy invariants of the considered knot K , therefore Floer homology theory is acategorication of the Alexander polynomial. For detail, see Khovanov homology Khovanov (2003).

5.8 Notes[1] Alexander describes his skein relation toward the end of his paper under the heading miscellaneous theorems, which is

possibly why it got lost. Joan Birman mentions in her paper New points of view in knot theory (Bull. Amer. Math. Soc.(N.S.) 28 (1993), no. 2, 253287) that Mark Kidwell brought her attention to Alexanders relation in 1970.

[2] Fintushel and Stern (1997) Knots, links, and 4-manifolds

5.9 References Alexander, J. W. (1928). Topological invariants of knots and links. Trans. Amer. Math. Soc. 30 (2):

275306. doi:10.2307/1989123. Crowell, R.; Fox, R. (1963). Introduction to Knot Theory. Ginn and Co. after 1977 Springer Verlag. Adams, Colin C. (2004). The Knot Book: An elementary introduction to the mathematical theory of knots

(Revised reprint of the 1994 original ed.). Providence, RI: American Mathematical Society. ISBN 0-8218-3678-1. (accessible introduction utilizing a skein relation approach)

Fox, R. (1961). A quick trip through knot theory, In Topology of ThreeManifold (Proceedings of 1961Topology Institute at Univ. of Georgia, edited by M.K.Fort ed.). Englewood Clis. N. J.: Prentice-Hall. pp.120167.

Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series 39.Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3.

Kauman, Louis (1983). Formal Knot Theory. Princeton University press. Kauman, Louis (2001). Knots and Physics. World Scientic Publishing Companey. Kawauchi, Akio (1996). A Survey of Knot Theory. Birkhauser. (covers several dierent approaches, explains

relations between dierent versions of the Alexander polynomial) Khovanov, M. (2006). Link homology and categorication. Proceedings of the ICM-2006. arXiv:math/0605339. Ozsvath, Peter; Szabo, Zoltan (2004). Holomorphic disks and knot invariants. Adv. Math., no., 58-6. Adv.

Math. 186 (1): 58116. arXiv:math/0209056. Bibcode:2002math......9056O. doi:10.1016/j.aim.2003.05.001.class=math.GT

Rasmussen, J. (2003). Floer homology and knot complements. PhD thesis Harvard University. p. 6378.arXiv:math/0306378. Bibcode:2003math......6378R.

Rolfsen, Dale (1990). Knots and Links (2nd ed.). Berkeley, CA: Publish or Perish. ISBN 0-914098-16-0.(explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)

5.10 External links Hazewinkel, Michiel, ed. (2001), Alexander invariants, Encyclopedia of Mathematics, Springer, ISBN 978-

1-55608-010-4 "Main Page" and "The Alexander-Conway Polynomial", The Knot Atlas. knot and link tables with computed

Alexander and Conway polynomials

Chapter 6

Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form

P = Q

where P and Q are polynomials with coecients in some eld, often the eld of the rational numbers. For mostauthors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, apolynomial equation may involve several variables, in which case it is called multivariate and the term polynomialequation is usually preferred to algebraic equation.For example,

x5 3x+ 1 = 0is an algebraic equation with integer coecients and

y4 +xy

2=

x3

3 xy2 + y2 1

7

is a multivariate polynomial equation over the rationals.Some but not all polynomial equations with rational coecients have a solution that is an algebraic expression witha nite number of operations involving just those coecients (that is, can be solved algebraically). This can be donefor all such equations of degree one, two, three, or four; but for degree ve or more it can only be done for someequations but not for all. A large amount of research has been devoted to compute eciently accurate approximationsof the real or complex solutions of an univariate algebraic equation (see Root-nding algorithm) and of the commonsolutions of several multivariate polynomial equations (see System of polynomial equations).

6.1 HistoryThe study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000BC could solve some kind of quadratic equations (displayed on Old Babylonian clay tablets).The algebraic equations over the rationals with only one variable are also called univariate equations. They have avery long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like x = 1+

p5

2for the positive solution of x2 x 1 = 0 . The ancient Egyptians knew how to solve equations of degree 2 inthis manner. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived thequadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant.During the Renaissance in 1545, Gerolamo Cardano found the solution to equations of degree 3 and Lodovico Ferrarisolved equations of degree 4. Finally Niels Henrik Abel proved, in 1824, that equations of degree 5 and equations ofhigher degree are not always solvable using radicals. Galois theory, named after variste Galois, was introduced togive criteria deciding if an equation is solvable using radicals.

14

6.2. AREAS OF STUDY 15

6.2 Areas of studyThe algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is thestudy of (univariate) algebraic equations over the rationals. Galois theory has been introduced by variste Galois forgetting criteria deciding if an algebraic equation may be solved in terms of radicals. In eld theory, an algebraic exten-sion is an extension such that every element is a root of an algebraic equation over the base eld. Transcendence theoryis the study of the real numbers which are not solutions to an algebraic equation over the rationals. A Diophantineequation is a (usually multivariate) polynomial equation with integer coecients for which one is interested in theinteger solutions. Algebraic geometry is the study of the solutions in an algebraically closed eld of multivariatepolynomial equations.Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalentwith P Q = 0 . It follows that the study of algebraic equations is equivalent to the study of polynomials.A polynomial equation over the rationals can always be converted to an equivalent one in which the coecients areintegers. For example, multiplying through by 42 = 237 and grouping its terms in the rst member, the previouslymentioned polynomial equation y4 + xy2 = x

3

3 xy2 + y2 17 becomes

42y4 + 21xy 14x3 + 42xy2 42y2 + 6 = 0:Because sine, exponentiation, and 1/T are not polynomial functions,

eTx2 +1

Txy + sin(T )z 2 = 0

is not a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomialequation in the three variables x, y, and z over the eld of the elementary functions in the variable T.As for any equation, the solutions of an equation are the values of the variables for which the equation is true. Forunivariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions ofthe algebraic equation P=0 are the roots of the polynomial P. When solving an equation, it is important to specify inwhich set the solutions are allowed. For example, for an equation over the rationals one may look for solutions in whichall the variables are integers. In this case the equation is a diophantine equation. One may also be interested only inthe real solutions. However, for univariate algebraic equations, the number of solutions is nite, and all solutions arecontained in any algebraically closed eld containing the coecientsfor example, the eld of complex numbers inthe case of equations over the rationals. It follows that without precision root and solution usually mean solutionin an algebraically closed eld.

6.3 See also Algebraic function Algebraic number Root nding Linear equation (degree = 1) Quadratic equation (degree = 2) Cubic equation (degree = 3) Quartic equation (degree = 4) Quintic equation (degree = 5) Sextic equation (degree = 6) Sept