Explicit formulas for the multivariate resultant

We present formulas for the homogenous multivariate resultant as a quotient of two determinants. They extend classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the given polynomials and coefficients of their Bezoutian. These formulas can also be viewed as an explicit computation of the morphisms and the determinant of a resultant complex.Comment: 30 pages, Late

Using projection operators in computer aided geometric design

We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention different formulations of resultants, including the projective resultant, the toric resultant, and the residual resultant. In the second part we illustrate these tools, and other projection operators, on typical problems of CAGD such as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface

On the Computation of Matrices of Traces and Radicals of Ideals

International audienceLet $f_1,\ldots,f_s \in \mathbb{K}[x_1,\ldots,x_m]$ be a system of polynomials generating a zero-dimensional ideal \I, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of ``matrices of traces" for the factor algebra \A := \CC[x_1, \ldots , 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 $\{M_{x_i}|i=1,\ldots,m\}$ of the radical \sqrt{\I}. We first propose a method using Macaulay type resultant matrices of $f_1,\ldots,f_s$ 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 \mathbb{P}^m_\CC. 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 $f_1,\ldots,f_m$ 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 \sqrt{\I} are given in terms of Bezoutians

Resultants and Discriminants for Bivariate Tensor-product Polynomials

International audienceOptimal resultant formulas have been systematically constructed mostly for unmixed polynomial systems, that is, systems of polynomials which all have the same support. However , such a condition is restrictive, since mixed systems of equations arise frequently in practical problems. We present a square, Koszul-type matrix expressing the resultant of arbitrary (mixed) bivariate tensor-product systems. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the entries of the matrix are simply coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. Moreover, for tensor-product systems with more than two (affine) variables, we prove an impossibility result: no universal degree-one formulas are possible, unless the system is unmixed. We present applications of the new construction in the computation of discriminants and mixed discriminants as well as in solving systems of bivariate polynomials with tensor-product structure