The RegularChains library in MAPLE

Abstract

Performing calculations modulo a set of relations is a basictechnique in algebra. For instance, computing the inverse of aninteger modulo a prime integer or computing the inverse of thecomplex number 3 + 2t modulo the relationℓ² + 1 = 0. Computing modulo a setS containing more than one relation requiresfrom S to have some mathematical structure. Forinstance, computing the inverse of p =x + y moduloS ={x² +y +1,y² +x + 1} is difficult unless one realizes thatthis question is equivalent to computing the inverse ofp modulo C ={x⁴ +2x² +x + 2,y +x² + 1}. Indeed, fromthere one can simplify p usingy =-x² - 1 leading toq =-x² +x - 1 and compute the inverse ofq modulox⁴ +2x² +x + 2 (using the extended Euclidean algorithm)leading to -1/2x³ -1/2x. One commonly used mathematical structurefor a set of algebraic relations is that of aGröbner basis. It is particularly wellsuited for deciding whether a quantity is null or not modulo a setof relations. For inverse computations, the notion of aregular chain is more adequate. For instance,computing the inverse of p =x + y modulo the setC ={y² -2x +1,x² -3x + 2}, which is both a Gröbner basisand a regular chain, is easily answered in this latter point ofview. Indeed, it naturally leads to consider the GCD ofp andCy =y² -2x + 1 modulo the relationCx =x² -3x + 2 = 0, which is [EQUATION] This shows that p has no inverse ifx = 1 and has an inverse (which can be computedand which is -y + 2) if x =2.