Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials (original) (raw)

Let A be a matrix whose entries are algebraic functions defined on a reduced quasiprojective algebraic set X , e.g., multivariate polynomials defined on X := C N . The sets S k (A), consisting of x ∈ X where the rank of the matrix function A(x) is at most k , arise in a variety of contexts. For example, in the description of both the singular locus of an algebraic set and its fine structure; in the description of the degeneracy locus of maps between algebraic sets; and in the computation of the irreducible decomposition of the support of coherent algebraic sheaves, e.g., supports of finite modules over polynomial rings. In this article we present a numerical algorithm to efficiently compute the sets S k (A).