Determinantal formulae for the solution set of zero-dimensional ideals

Abstract

In this paper we study the properties of a forgotten construction introduced by V.W. Habicht in 1948 related with the problem of computing a basis for the ideal generated by n polynomials F 1,…, F n in K[x n,…,x n allowing to determine in an easy way its zeros and its radical for the zero-dimensional case. This construction is achieved generalizing to K[x 1,…,x n the definition of subresultants of indexes 0 and 1 and so, the coefficients of the polynomials in the new basis are defined as the determinants of certain matrices constructed with the coefficients of the polynomials F 1,…, F n . The construction is introduced for the generic case and it is showed that its specialization works for all the zero-dimensional ideals in general position with respect to x n except when the coefficients of the polynomials F 1,…, F n are in some hypersurface in K n whose equation we shall give in an explicit way. A basis with a similar structure to the one here introduced can be computed using Groebner Basis (the Shape Lemma) imposing the radicality hypothesis to the initial ideal.