The RegularChains library in MAPLE

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 + 2<i>t</i> modulo the relation&ell;² + 1 = 0. Computing modulo a set<i>S</i> containing more than one relation requiresfrom <i>S</i> to have some mathematical structure. Forinstance, computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo<i>S</i> ={<i>x</i>² +<i>y</i> +1,<i>y</i>² +<i>x</i> + 1} is difficult unless one realizes thatthis question is equivalent to computing the inverse of<i>p</i> modulo <i>C</i> ={<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2,<i>y</i> +<i>x</i>² + 1}. Indeed, fromthere one can simplify <i>p</i> using<i>y</i> =-<i>x</i>² - 1 leading to<i>q</i> =-<i>x</i>² +<i>x</i> - 1 and compute the inverse of<i>q</i> modulo<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2 (using the extended Euclidean algorithm)leading to -1/2<i>x</i>³ -1/2<i>x</i>. One commonly used mathematical structurefor a set of algebraic relations is that of a<i>Gr&ouml;bner basis.</i> It is particularly wellsuited for deciding whether a quantity is null or not modulo a setof relations. For inverse computations, the notion of a<i>regular chain</i> is more adequate. For instance,computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo the set<i>C</i> ={<i>y</i>² -2<i>x</i> +1,<i>x</i>² -3<i>x</i> + 2}, which is both a Gr&ouml;bner basisand a regular chain, is easily answered in this latter point ofview. Indeed, it naturally leads to consider the GCD of<i>p</i> and<i>C<inf>y</inf></i> =<i>y</i>² -2<i>x</i> + 1 modulo the relation<i>C<inf>x</inf></i> =<i>x</i>² -3<i>x</i> + 2 = 0, which is [EQUATION] This shows that <i>p</i> has no inverse if<i>x</i> = 1 and has an inverse (which can be computedand which is -<i>y</i> + 2) if <i>x</i> =2.