17 research outputs found
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms
We develop a theory of Tannakian Galois groups for t-motives and relate this
to the theory of Frobenius semilinear difference equations. We show that the
transcendence degree of the period matrix associated to a given t-motive is
equal to the dimension of its Galois group. Using this result we prove that
Carlitz logarithms of algebraic functions that are linearly independent over
the rational function field are algebraically independent.Comment: 39 page
Zeros of analytic functions, with or without multiplicities
The classical Mason-Stothers theorem deals with nontrivial polynomial
solutions to the equation . It provides a lower bound on the number of
distinct zeros of the polynomial in terms of the degrees of , and
. We extend this to general analytic functions living on a reasonable
bounded domain , rather than on the whole of . The estimates obtained are sharp, for any , and a generalization of
the original result on polynomials can be recovered from them by a limiting
argument.Comment: This is a retitled and slightly revised version of my paper
arXiv:1004.359
Zero estimates on group varieties II
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46615/1/222_2005_Article_BF01388605.pd