Fast Probabilistic Algorithms for Verification of Polynomial Identities

Abstract

The starthng success of the Rabm-Strassen-Solovay pnmahty algorithm, together with the intriguing foundattonal posstbthty that axtoms of randomness may constttute a useful fundamental source of mathemaucal truth independent of the standard axmmaUc structure of mathemaUcs, suggests a wgorous search for probabdisuc algonthms In dlustratmn of this observaUon, vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials. Ancdlary fast algorithms for calculating resultants and Sturm sequences are given. Probabilistlc calculatton in real anthmetlc, prewously considered by Davis, is justified ngorously, but only in a special case. Theorems of elementary geometry can be proved much more efficiently by the techmques presented than by any known arttficml-mtelhgence approach KEY WORDS AND PHRASES: polynommls, polynomml algonthms, probabdtsuc algorithms CR CATEGORIES. 5.21, 5.7 Integer Probabilistic Calculations for Multivariate PolynomialsThe startling success of the Rabin-Strassen-Solovay algorithm (see Rabin [17]), together with the intriguing foundational possibility that axioms of randomness may constitute a useful fundamental source of mathematical truth independent of, but supplementary to, the standard axiomatic structure of mathematics (see Chaitin and Schwartz [3l), suggests that probabilistic algorithms ought to be sought vigorously. As an illustration of what may be possible, this paper presents probabilistic algorithms for testing asserted multivariable polynomial identities Q = R, as well as other asserted or conjectured relationships between sets of polynomials, e.g., the assertion that one polynomial Q belongs to the ideal generated by finitely many others.The technique that we use is essentially elementary. Given a purported polynomial identity, we can always write it as Q ffi 0. We do not suppose that the Q presented to us for testing is given in standard simplified polynomial form. For example, if we did not immediately recognize its truth, we might wish to test the identity (x + y)(x -y) -x 2 + y2 = 0. Indeed, if we write Q for the standard simplified form of Q, what we want is precisely a test to determine whether all the coefficients of Q are zero.We allow our polynomials to have coefficients in any field or integral domain F. At some points in our argument the condition that F should be infinite will play an essential role. We write deg(Q) for the degree of Q and [ S[ for the cardinality of a set S.Note that it will generally be trivial to develop upper bounds for deg(Q) directly from the expression structure of Q.

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