Solving determinantal systems using homotopy techniques

02/28/2018

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by   Jonathan D. Hauenstein, et al.

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Let be a field of characteristic zero and be an algebraic closure of . Consider a sequence of polynomialsG=(g_1,...,g_s) in [X_1,...,X_n], a polynomial matrix =[f_i,j] ∈[X_1,...,X_n]^p × q, with p ≤ q,and the algebraic set V_p(F, G) of points in at which all polynomials in and all p-minors of 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 . 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.