112 research outputs found
Implicitization of Bihomogeneous Parametrizations of Algebraic Surfaces via Linear Syzygies
We show that the implicit equation of a surface in 3-dimensional projective
space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a
given positive integer d, can be represented and computed from the linear
syzygies of its parametrization if the base points are isolated and form
locally a complete intersection
Resultant of an equivariant polynomial system with respect to the symmetric group
Given a system of n homogeneous polynomials in n variables which is
equivariant with respect to the canonical actions of the symmetric group of n
symbols on the variables and on the polynomials, it is proved that its
resultant can be decomposed into a product of several smaller resultants that
are given in terms of some divided differences. As an application, we obtain a
decomposition formula for the discriminant of a multivariate homogeneous
symmetric polynomial
On the irreducibility of multivariate subresultants
Let be generic homogeneous polynomials in variables of
degrees respectively. We prove that if is an integer
satisfying then all multivariate
subresultants associated to the family in degree are
irreducible. We show that the lower bound is sharp. As a byproduct, we get a
formula for computing the residual resultant of
smooth isolated points in \PP^{n-1}.Comment: Updated version, 4 pages, to appear in CRA
Résultant univarié et courbes algébriques planes
MasterLe premier chapitre traite du résultant de Sylvester qui constitue l'outil essentiel de ce cours. Le deuxième chapitre propose une étude effective du problème de l'intersection de deux courbes algébriques planes: théorème de Bézout, notion de multiplicité d'intersection et calcul de points d'intersection par valeurs et vecteurs propres. Le troisième chapitre aborde la manipulation des courbes algébriques planes rationnelles: degré d'une paramétrisation, problèmes d'implicitation et d'inversion d'une paramétrisation. Enfin, quelques compléments sont donnés sous forme d'exercices dans le dernier chapitre
On the equations of the moving curve ideal of a rational algebraic plane curve
Given a parametrization of a rational plane algebraic curve C, some explicit
adjoint pencils on C are described in terms of determinants. Moreover, some
generators of the Rees algebra associated to this parametrization are
presented. The main ingredient developed in this paper is a detailed study of
the elimination ideal of two homogeneous polynomials in two homogeneous
variables that form a regular sequence.Comment: Journal of Algebra (2009
Formulas for the eigendiscriminants of ternary and quaternary forms
A -dimensional tensor of format
defines naturally a rational map from the projective space
to itself and its eigenscheme is then the subscheme of
of fixed points of . The eigendiscriminant is an
irreducible polynomial in the coefficients of that vanishes for a given
tensor if and only its eigenscheme is singular. In this paper we contribute two
formulas for the computation of eigendiscriminants in the cases and
. In particular, by restriction to symmetric tensors, we obtain closed
formulas for the eigendiscriminants of plane curves and surfaces in
as the ratio of some determinants of resultant matrices
Extraction of cylinders and cones from minimal point sets
We propose new algebraic methods for extracting cylinders and cones from
minimal point sets, including oriented points. More precisely, we are
interested in computing efficiently cylinders through a set of three points,
one of them being oriented, or through a set of five simple points. We are also
interested in computing efficiently cones through a set of two oriented points,
through a set of four points, one of them being oriented, or through a set of
six points. For these different interpolation problems, we give optimal bounds
on the number of solutions. Moreover, we describe algebraic methods targeted to
solve these problems efficiently
Elimination and nonlinear equations of Rees algebra
A new approach is established to computing the image of a rational map,
whereby the use of approximation complexes is complemented with a detailed
analysis of the torsion of the symmetric algebra in certain degrees. In the
case the map is everywhere defined this analysis provides free resolutions of
graded parts of the Rees algebra of the base ideal in degrees where it does not
coincide with the corresponding symmetric algebra. A surprising fact is that
the torsion in those degrees only contributes to the first free module in the
resolution of the symmetric algebra modulo torsion. An additional point is that
this contribution -- which of course corresponds to non linear equations of the
Rees algebra -- can be described in these degrees in terms of non Koszul
syzygies via certain upgrading maps in the vein of the ones introduced earlier
by J. Herzog, the third named author and W. Vasconcelos. As a measure of the
reach of this torsion analysis we could say that, in the case of a general
everywhere defined map, half of the degrees where the torsion does not vanish
are understood
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