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Let A be a matrix whose entries are algebraic functions defined on a reduced quasi-projective algebraic set X, e.g. multivariate polynomials defined on X:= ℂ N . The sets  Sk(A){\mathcal 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 compute the sets  Sk(A){\mathcal S}_k(A) efficiently.

Numerical Decomposition of the Rank-Deficiency Set of a Matrix of Multivariate Polynomials