Solving determinantal systems using homotopy techniques

Abstract

Let K be a field of characteristic zero and K be an algebraic closure of K. Consider a sequence of polynomials G = (g 1 ,. .. , g s) in K[X 1 ,. .. , X n ], a polynomial matrix F = [f i,j ] ∈ K[X 1 ,. .. , X n ] p×q , with p ≤ q, and the algebraic set V p (F, G) of points in K at which all polynomials in G and all p-minors of F vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry. We provide bounds on the number of isolated points in V p (F, G) depending on the maxima of the degrees in rows (resp. columns) of F. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining V p (F, G). In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

Homotopy techniques for solving sparse column support determinantal polynomial systems

“…Proposition 3.1 below makes these requirements more precise; it is a minor modification of [17,Propositions 13 and 24]. To state it, it will be convenient to describe our homotopy process using only vectors of polynomials.…”

“…Finally, as in [17], the following proposition assumes that we are given a straight-line program Γ that computes the polynomials B, that is, is a sequence of operations +, −, × that takes as input t, x 1 , . .…”

“…Such algorithms first appeared in [7,19] without any structure on the system. Later symbolic homotopies were used in square sparse systems [25,20,21,22] and multi-homogeneous systems [30,18,17].…”

“…Then, given a zero-dimensional parametrization R 0 of V p (M , r), we will apply the algorithm in [17] to the system (M p (V ), u) to lift R 0 to a zero-dimensional parametrization R 1 of the isolated zeros of V p (F , g). At a high level the strategy for using homotopy methods to determine isolated zeros is relatively simple to describe, but also difficult to realize.…”

“…This is not the first time that determinantal structures have been exploited to speed-up polynomial system solvers. Previous work includes, for example, [17], which is also based on homotopy techniques: we borrow some results and techniques from that reference, but our discussion of the "sparse" aspects is new. Note also that one can encode rank deficiencies in a polynomial matrix using extra variables (sometimes called Lagrange multipliers in the context of polynomial optimization) to encode that the kernel of the considered matrix is non-trivial.…”