Homotopy techniques for solving sparse column support determinantal polynomial systems

• Published in: Journal of Complexity Vol. 66; p. 101557
• Main Authors: Labahn, George, Safey El Din, Mohab, Schost, Éric, Vu, Thi Xuan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-10-2021; Elsevier
• Subjects: Computer Science; Determinantal systems; Sparse polynomials; Symbolic Computation; Symbolic homotopy; Sparse polynomials; Determinantal systems; Symbolic homotopy
• ISSN: 0885-064X; 1090-2708
• DOI: 10.1016/j.jco.2021.101557

Let K be a field of characteristic zero with K¯ its algebraic closure. Given a sequence of polynomials g=(g1,…,gs)∈K[x1,…,xn]s and a polynomial matrix F=[fi,j]∈K[x1,…,xn]p×q, with p≤q, we are interested in determining the isolated points of Vp(F,g), the algebraic set of points in K¯ at which all polynomials in g and all p-minors of F vanish, under the assumption n=q−p+s+1. Such polynomial systems arise in a variety of applications including for example polynomial optimization and real algebraic geometry.
We design a randomized sparse homotopy algorithm for computing the isolated points in Vp(F,g) which takes advantage of the determinantal structure of the system defining Vp(F,g). Its complexity is polynomial in the maximum number of isolated solutions to such systems sharing the same sparsity pattern and in some combinatorial quantities attached to the structure of such systems. It is the first algorithm which takes advantage both on the determinantal structure and sparsity of input polynomials.
We also derive complexity bounds for the particular but important case where g and the columns of F satisfy weighted degree constraints. Such systems arise naturally in the computation of critical points of maps restricted to algebraic sets when both are invariant by the action of the symmetric group.