Solving determinantal systems using homotopy techniques

Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g_1,\dots,g_s)$ in $\K[X_1,\dots,X_n]$, a polynomial matrix $\F=[f_{i,j}] \in \K[X_1,\dots,X_n]^{p \times q}$, with $p \leq q$,and the algebraic set $V_p(F, G)$ of points in $\KKbar$ 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.

💡 Research Summary

The paper addresses the computational problem of solving over‑determined polynomial systems that are defined by the vanishing of all p‑minors of a p × q matrix F (with p ≤ q) together with a set of auxiliary polynomials G = (g₁,…,g_s). Such systems naturally arise in polynomial optimization (Jacobian matrices plus Lagrange multipliers), computational geometry and real algebraic geometry.

Problem setting.
Given a field K of characteristic zero, its algebraic closure K̅, variables X₁,…,Xₙ, a matrix F ∈ K