Some combinatorial-algebraic problems from complexity theory
Discrete Mathematics - Special issue: trends in discrete mathematics
Reducing randomness via irrational numbers
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Simplified lower bounds for polynomials with algebraic coefficients
Journal of Complexity
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Information Processing Letters
Modern computer algebra
On the rigidity of Vandermonde matrices
Theoretical Computer Science
On testing for zero polynomials by a set of points with bounded precision
Theoretical Computer Science - Computing and combinatorics
Algebraic Complexity Theory
Rigidity of a simple extended lower triangular matrix
Information Processing Letters
Complexity Lower Bounds using Linear Algebra
Foundations and Trends® in Theoretical Computer Science
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The rigidity of a matrix A with respect to the rank bound r is the minimum number of entries of A that must be changed to reduce the rank of A to or below r. It is a major unsolved problem (Valiant, 1977) to construct “explicit” families of n × n matrices of rigidity n1+δ for r=εn, where ε and δ are positive constants. In fact, no superlinear lower bounds are known for explicit families of matrices for rank bound r=Ω(n). In this paper we give the first optimal, Ω(n2), lower bound on the rigidity of two “somewhat explicit” families of matrices with respect to the rank bound r=cn, where c is an absolute positive constant. The entries of these matrix families are (i) square roots of n2 distinct primes and (ii) primitive roots of unity of prime orders for the first n2 primes. Our proofs use an algebraic dimension concept introduced by Shoup and Smolensky (1997) and a generalization of that concept.