872 research outputs found
Tropicalization and irreducibility of Generalized Vandermonde Determinants
We find geometric and arithmetic conditions in order to characterize the
irreducibility of the determinant of the generic Vandermonde matrix over the
algebraic closure of any field k. We also characterize those determinants whose
tropicalization with respect to the variables of a row is irreducible.Comment: 10 pages, AMSart. Revised version to appear in Proceedings of the AM
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
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
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