The RegularChains library in MAPLE

• Authors: F. Lemaire;M. Moreno Maza;Y. Xie
• Venue: ACM SIGSAM Bulletin
• Year: 2005

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.