Journal of Pure and Applied Algebra 76 ( I991 ) 57-80 North-Holland 57 eter ant zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB ideals Law-eano Gonz5lez-Vega” Departnmento Matemriticcrs. Estudisticn y Comptmcidtt . Fmrhd de Ciertcius. Uttiversidud de Cnnthia. Smtctttder 39071. Sppctitt Communicated by M.F. Coste-Roy Received Y November IYYO Revised 25 January 1991 A Ostract Gonzilez-Vega. L.. Determinantal formulae for the solution set of zero-dimensional ideals. Journal of Pure and Applied Algebra 76 ( IYY I) 57-80. 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 tl polynomials F,. . . . . F,, in W[s,. . . . . s,,] allowing to determine in an easy way its zeros and its radical for the zero-dimensional case. This construction is achieved gent-r&zing to Dd!s,. . . . .x,,] 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,, . . . F,, . 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,, except when the coefficients of the polynomials F,. . . . F,, are in some hypersurface in K” 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. zyxwvutsrqponmlkjihgfedcbaZYX Introduction Let IL be a field of characteristic 0 and K its algebraic closure. zyxwvutsrqponmlkjihgfedcbaZYXWVUT Let zyxwvutsrqponmlkjihgfedcbaZY F,. . . . . F,,, be m polynomials in Qu] = [L[x,, . . . ,.Y,,] such that: l the ideal generated by F,, . . . , F,,, in K[x,, . . . ,x,,] is zero-dimensional, i.e.+ card@EK”1 F,(g)=0 . . . . . F,,,(c~)=o))<+~, * Partially supported wy ClCyT XYiO37YicOZ/Ol. 0022-4049/91/$03.50 0 lY91 - Elsevier Science Publishers B.V. All rights rcscrved