Fast Probabilistic Algorithms for Verification of Polynomial Identities

The s tar thng success o f the Rabm-S t ra s sen -So lovay p n m a h t y algori thm, together wi th the intr iguing foundat tonal posstbthty that axtoms of randomness may constttute a useful fundamenta l source o f m a t h e m a u c a l 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