The Fundamental Theorem of Algebra RevisitedAuthor(s): Airton von Sohsten de MedeirosSource: The American Mathematical Monthly, Vol. 108, No. 8 (Oct., 2001), pp. 759-760Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2695621 .

Accessed: 13/04/2014 09:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

http://www.jstor.org

This content downloaded from 81.241.31.4 on Sun, 13 Apr 2014 09:30:16 AMAll use subject to JSTOR Terms and Conditions

Katz and Tao suggest in [3] that it may be possible to improve the bound in Theo- rem 3 by including more conditions of the form

I{a + kb : (a, b) E G} < N.

This corresponds to more slices of the Kakeya set and would enable an improvement of our bound with little further trouble.

ACKNOWLEDGEMENT. Thanks to A. Carbery and W. T. Gowers for all their invaluable help.

REFERENCES

1. A. S. Besicovitch, The Kakeya Problem, Amer. Math. Monthly 70 (1963) 697-706. 2. J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9

(1999) 256-282. 3. N. H. Katz and T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture,

preprint. 4. T. Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995)

651-674. 5. T. Wolff, Recent work connected with the Kakeya problem, in Prospects in Mathematics: Invited Talks on

the Occasion of the 250th Aniversity of Princeton University March 17-21, 1996 Princeton University, Hugo Rossi, ed., American Mathematical Society, Providence, RI, 1998, pp. 129-162.

Cambridge University, DPMMS, 16 Mill Lane, Cambridge, CB2 JSB, U.K. kmrogers.123@hotmail.com

The Fundamental Theorem of Algebra Revisited

Airton von Sohsten de Medeiros

We present a proof based upon the Lefschetz Fixed-point Theorem of the

Fundamental Theorem of Algebra. Every nonconstant polynomial with complex coefficients has a (complex) zero.

We recall the

Lefschetz Fixed-Point Theorem. Let X be a compact polyhedron and let f: X-+ X be a continuous mapping. If the Lefschetz number of f is not zero, then f has a fixed point.

All we need to know about the Lefschetz number is that homotopic mappings have the same Lefschetz number and that, for the identity mapping, this number coincides with the Euler characteristic of the polyhedron [2, pp. 194-195].

The main idea, which allows us to make use of the Lefschetz Theorem, is that a linear isomorphism A of the complex n-space C'n induces (in the obvious way) a continuous mapping A on the complex projective (n - 1)-space CP(n - 1), viewed

October 2001] NOTES 759

This content downloaded from 81.241.31.4 on Sun, 13 Apr 2014 09:30:16 AMAll use subject to JSTOR Terms and Conditions

as the set of all complex straight lines through the origin of C'(. The fixed points of this mapping (if there are any) are exactly the A-invariant 1-dimensional subspaces. Hence, the characteristic polynomial of A has a zero if and only if A has a fixed point.

Now, given the polynomial p(z) of degree n > 1, which we may assume to be monic, let A be any linear operator on C/n whose characteristic polynomial is precisely p(z) [1, pp. 69-70] and consider the linear homotopy A, = tA + (1 - t)I, where t E [0, 1] and I denotes the identity mapping on Cn.

If det A, = 0 for some t, then t A 0 and we are done, for 1- t1 turns out to be a zero of p(z). Otherwise, A, is a linear isomorphism for all values of t, inducing thus, the homotopy A, between A and the identity mapping on CP(n - 1). In particular, these two mappings have the same Lefschetz number X = n (the Euler characteristic of C P(n - 1)). This ensures the existence of a fixed point of A and, consequently, of a zero of p(z).

REFERENCES

1. William C. Brown, Matrices over Commutative Rings, Marcel Dekker, Inc., New York, 1993. 2. Edwin H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1966.

Instituto de Matemdtica, UFRJ, Departamento de Matemdtica Aplicada, Ilha do Funddo, Rio de Janeiro, Brazil airton@ impa.br

Other Versions of the Steiner-Lehmus Theorem

Mowaffaq Hajja

The Steiner-Lehmus Theorem Inside-out was the name given in [1] to the following theorem:

Theorem 1. If BB' and CC' are the cevians of the acute-angled triangle ABC through its circumcenter 0 (Figure 1) and if AB' = AC', then AB = AC.

A

IB

Figure 1.

760 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

This content downloaded from 81.241.31.4 on Sun, 13 Apr 2014 09:30:16 AMAll use subject to JSTOR Terms and Conditions