On the rigidity of Vandermonde matrices

Abstract

The rigidity function R A(r) of a matrix A is the minimum number of entries of A that must be changed to reduce the rank of A to less than or equal to r. While almost all matrices have rigidity close to (n−r) 2 , proving strong lower bounds on the rigidity of explicit matrices is a fundamental open question with several consequences in complexity theory. A natural class of matrices expected to have high rigidity is that of Vandermonde matrices V=(x i j−1) 1⩽i, j⩽n . However, even when the x i are algebraically independent, it was not known if R V(r)= Ω(n 2) for nonconstant r. We prove that for any constant c<1, there exists a constant ε>0 such that if r⩽ε n , then R V(r)⩾cn 2 , when the x i are algebraically independent. Although not explicit, this provides a natural n-dimensional manifold in the space of n×n matrices with Ω(n 2) rigidity for nonconstant r. Our proof is based on a technique due to Shoup and Smolensky (Comput. Complexity 6(4) (1997) 301–311). For explicit Vandermonde matrices, the best-known lower bound is R V(r)= Ω(n 2/r log(n/r)) , when log 2 n⩽r⩽n/2 , which follows from a result of Shokrollahi et al. (Inform. Process. Lett. 64(6) (1997) 283–285).