Solving zero-dimensional involutive systems
A.Yu. Zharkov · Birkhäuser Basel eBooks · 1996
In our recent paper [1], the notion of involutive bases of polynomial ideals was introduced and an algorithm for computing involutive bases was presented. The improved form of this algorithm together with the proof of its correctness in the zero-dimensional case is given in [2]. In the positive-dimensional case, a linear change of variables is generally required for constructing involutive bases defined in our sense. It turns out that when the involutive basis exists (without change of variables), it can be computed considerably faster by our algorithm than the minimal standard basis by Buchberger’s algorithm [3]. On the other hand, an involutive basis computed in the total-degree term ordering often looks more complicated than the corresponding minimal standard basis. The reason is that the involutive basis of a zero-dimensional ideal is nothing but a standard basis enlarged to an “overdetermined” linear algebraic system in monomials irreducible modulo this ideal. From this fact, some interesting properties of involutive bases may be deduced, and a simple method for solving zero-dimensional systems may be constructed.