A subresultant theory for multivariate polynomials

In Computer Algebra, Subresultant Theory provides a powerful method to construct algorithms solving problems for polynomials in one variable in an optimal way. For instance, using this method, we can compute the greatest common divisor of two polynomials in one variable with integer coefficients avoiding the exponential growth of the coefficients that will appear if we use the Euclidean Algorithm. In this paper, generalizing a forgotten construction appearing in [Hab], we extend the Subresultant Theory to the multivariate case. In order to achieve this, first of all, we introduce the definition of subresultant sequence asociated to two polynomials in one variable with coefficients in an integral domain, describing the properties of this sequence that we would like to extend to the multivariate case. In the second section we generalize the definition of subresultant polynomial to the multivariate case, showing that many of the properties obtained in the one variable case work also in the multivariate case. In this way we show how these subresultants can be used to get a greatest common divisor of n polynomialsinll[zl,. ... Zn–l] where D is an integral domain. In the third section we apply this Subresultant Theory to get a determinantal formula for the solution set of almost all O-dimensional ideals defined by n polynomials in ll[zl, . . . . X,l], with D an integral domain. The formula, in some sense generalizes Cramer’s rule for solving linear systems of equations. Finally, in the last section, we show some open problems related with this construction.