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TitleOn the computation of matrices of traces and radicals of ideals
Author(s) Itnuit Janovitz-Freireich, Bernard Mourrain, Lajos Rónyai, Ágnes Szántó
TypeArticle in Journal
AbstractLet f 1 , … , f s ∈ K [ x_1 , … , x_m ] be a system of polynomials generating a zero-dimensional ideal I , where K is an arbitrary algebraically closed field. We study the computation of “matrices of traces” for the factor algebra A ≔ K [ x_1 , … , x_m ] / I , i.e. matrices with entries which are trace functions of the roots of I . Such matrices of traces in turn allow us to compute a system of multiplication matrices M x i ∣ i = 1 , … , m of the radical I . We first propose a method using Macaulay type resultant matrices of f 1 , … , f s and a polynomial J to compute moment matrices, and in particular matrices of traces for A . Here J is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when I has finitely many projective roots in P K m . We also extend previous results which work only for the case where A is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of A . Here we need the assumption that s = m and f 1 , … , f m define an affine complete intersection. This second method also works if we have higher-dimensional components at infinity. A new explicit description of the generators of I are given in terms of Bezoutians.
KeywordsMatrix of traces, Radical of an ideal
URL http://www.sciencedirect.com/science/article/pii/S0747717111001325
JournalJournal of Symbolic Computation
Pages102 - 122
Translation No
Refereed No