A Note on Abnormal Polynomial Remainder Sequences

S. C. Johnson

Bell Laboratories, Murray Hill, New Jersey 07974

A B S T R A C T

Polynomial Remainder Sequenq~ (PRS) have been used in tb,~ computa- tion of greatest common divisors of polynomials for many years. We'call a po- lynomial division abnormal if the degree of the remainder is strictly less than the degree of the divisor minus one; an abnormal PRS is one with at least one abnornial division. An abnormal polynomial division in general implies a non- trivial greatest common divisor among the coefficients of the remainder; this can lead to inefficiencies in the later computat ions in the PRS.

These[re'suits suggest that the "classical" PRS methods may have not ex- ploited all o] th~a.tgebraic structure available in the problem.

Polynomial Remainder S e q u e n c e s (PRS) have been widely used tq, compute Greatest Common Divisors (gcd~s) of poly- nomials (See [11, [2], and [31). This note makes an observation about abnormal poly- nomial divisions which is relevant to abnor- mal PRS. We use a number of facts about the division of polynomials; the proofs can be deduced by the methods of [2] or [3].

Let A and B be polynomials in x over a unique factorization domain D. Suppose that the x degree of A, d ( A ) , is not less than d(B) . Then there exists an element a of D, and polynomials Q and C, such that

and

c,A =QB+ C

d ( C ) < d ( B )

d ( Q ) = d ( A ) - - d ( B )

The ratios Clot and Q/a over the quotient field of D are independent of the particular o~, C, and Q chosen.

Let us say d ( A ) = n , d ( B ) = l n , with mR<n, and set

= 1 1 - - m

A =ao xn + a I x n - I + . . . + a,,

B =bo xm + b I x m - I + . . . +bm

Then, we can choose

a =b~ + 1

as is traditionally done in the reduced and subresultant PRS algorithms. In this case, the division is called a pseudodivision, and Q and C are called the pseudoquotient and pseu- doremainder, respectively.

If, in fact, some smaller value of a would allow, the division to proceed in D[x] , this implies 'that the computed value of C is nonprimitive; this is a bad situation, as the nonprimitivity tends to grow later in the PRS.

The above choice of a is too large when a 0 and b 0 have a nontrivial gcd, g. Then, in fact,

o~ = b~ + 1 / g

is sufficient to allow division in D[x].

A less well known case where the above o~ overestimates is when 8 > O, and b 0 and b I h a v e a nontr ivialgcd, h. Then,

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is sumcient to allow division in D[x].

We shall call a division abnormal if d ( C ) < m - l . (This generally agrees with the usual definition.) The main result of this note is

Theorem." If the pseudodivision

aA =QB+ C

is abnormal then ei ther bola 0 or gcd(b O, b 1 ) is nontrivial.

Proof" Suppose gcd(bo, b I) = 1. Then we shall show that b0]a 0 by induction on 8.

When 8 = 0, the coefficient of x m-I in b o A - a o B is boa t - a o b I. By abnormali- ty, this is0. Thus b 0 divides aobl, and thus divides a 0.

When 8 > 0, assume the result is true f o r S - l . Let E=boA--aoxSB. Then if the division of A by B is abnormal, so is the division of E by B. By the induction hy- pothesis, if gcd (b 0, b I ) = 1 then b 0 divides the leading coefficient of E. The coefficient o f x " -1 in E i s boa I - a o b I. Either this i s0 or it is the leading coefficient; in ei ther case, b 0 divides it. Thus, b 0 divides aobl, so b 0 divides a0, completing the induction.

The above theorem implies that, in an abnormal pseudodi'vision, if B is not monic and 8 > 0, then C must be nonprinlitive, and must in fact have some nontrivial divi- sor in common with b 0. In practice, in a PRS most pseudoremainders are actually nonprimitive, even when the divisions are normal (this is the whole point of the re- duced and subresultant PRS algorithms!). The only immediate application of the above theorem would seem to be when the first division in a PRS is abnormal.

Nevertheless, this result is a bit surprising, and suggests that there may still be algebraic structure undiscovered in the gcd problem. Perhaps the most persuasive message is that the "classical" PRS methods are still not completely understood.

REFERENCES

[1] Brown, W. S., "On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors", J.A.C.M. 18 (1971) p.p. 478-504~

[2] Brown, W. S., and Traub, J. F., "On Euclid's Algorithm and the Theory of Subresultants", J.A.C.M., 18 (1971) p.p. 5-05-514.

[3] Collins, G. E., "Subresultants and Re- duced Polynomial Remainder Se- quences", J.A.C.M., 19 (1967) p.p. 128-142.

(continued from page 20)

References

[AHU 74] Alfred V. Aho, John E. Hopcroft, and Jeffery D. Ullman, The Design and Analysis of Computer Al- gorithms, Addison-Wesley Publishing Co., Reading, Mass., 1974

[FITC 74] John Fitch, A Simple Method of Taking nth Roots of Integers, SIGSAM Bulletin, Vol. 8, No. 4 (Nov. 1974), p. 26.

S. C../oh.so.

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