Using Elimination Theory to construct Rigid Matrices

Abstract

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
\cite{Val77}, rigidity and similar rank-robustness functions of
matrices have found numerous applications in circuit complexity,
communication complexity, and learning complexity. Almost all $\nbyn$
matrices over an infinite field have a rigidity of $(n-r)^2$. It is a
long-standing open question to construct infinite families of
\emph{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 $\nbyn$ matrix in this family are distinct primitive roots of unity
of orders roughly \SL{$\exp(n^4 \log n)$}. 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 semicontinuous.