arXiv subjects

Agnes Szanto

Publications and source records attributed to Agnes Szanto.

25 records · Page 2Linked to original sources

arXiv2009

On the Computation of Matrices of Traces and Radicals of Ideals

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ 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 $\A := \CC[x_1, ..., x_m]/ \I$, 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 $\sqrt{\I}$. 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 $\mathbb{P}^m_\CC$. 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 $f_1,...,f_m$ 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.

cs.SC↗

arXiv2008

Moment matrices, trace matrices and the radical of ideals

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. Assume that the factor algebra $\A=\mathbb{K}[x_1,...,x_m]/\I$ is Gorenstein and that we have a bound $δ>0$ such that a basis for $\A$ can be computed from multiples of $f_1,...,f_s$ of degrees at most $δ$. We propose a method using Sylvester or Macaulay type resultant matrices of $f_1,...,f_s$ and $J$, where $J$ is a polynomial of degree $δ$ generalizing the Jacobian, to compute moment matrices, and in particular matrices of traces for $\A$. These matrices of traces in turn allow us to compute a system of multiplication matrices $\{M_{x_i}|i=1,...,m\}$ of the radical $\sqrt{\I}$, following the approach in the previous work by Janovitz-Freireich, Rónyai and Szántó. Additionally, we give bounds for $δ$ for the case when $\I$ has finitely many projective roots in $\mathbb{P}^m_\CC$.

cs.SC↗

arXiv2008

A Bound for Orders in Differential Nullstellensatz

We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz.

math.AC↗

arXiv2008

Sylvester's Double Sums: the general case

In 1853 Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to charactise all members of the family, providing a uniform method.

math.AC↗

arXiv2007

Approximate radical for clusters: a global approach using Gaussian elimination or SVD

We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in C[x1, . . ., xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of I. To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if I has k distinct zero clusters each of radius at most epsilon in the max-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically bounded by epsilon^2. We also show that the (k+1)-th singular value of the matrix of traces is O(epsilon^2). The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically bounded by epsilon^2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.

math.AG↗

arXiv2006

An Elementary Proof of Sylvester's Double Sums for Subresultants

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.

math.AC↗

arXiv2005

Multivariate Subresultants in Roots

We give rational expressions for the subresultants of n+1 generic polynomials f_1,..., f_{n+1} in n variables as a function of the coordinates of the common roots of f_1,..., f_n and their evaluation in f_{n+1}. We present a simple technique to prove our results, giving new proofs and generalizing the classical Poisson product formula for the projective resultant, as well as the expressions of Hong for univariate subresultants in roots.

math.AG↗