On deflation and multiplicity structure

Abstract

International audienceThis paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are " exact " in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

“…Second, based on our work [35] and the work of Mourrain et al [21,36], as an application of our method, we give a heuristic method for certifying not only the isolated singular zeros of polynomial systems, but also the multiplicity structures of the isolated singular zeros of polynomial systems.…”

“…In recent years, Mourrain et al [21,36] proposed a new deflation method, which can be used to refine the accuracy of an isolated singular zero and the parameters introduced simultaneously and, moreover, the parameters can describe the multiplicity structure at the zero. They also proved that the number of equations and variables in this deflation method depend polynomially on the number of variables and equations of the input system and the multiplicity of the singular zero.…”

TEMPORARY REMOVAL: On isolation of singular zeros of multivariate analytic systems

“…[MM11] proposed a one-step deflation method to verify a multiple zero of a nearby system with a given multiplicity structure, which depends on the accuracy of the given approximate multiple zero. [HMS17] proposed a novel deflation method to verify the existence of an isolated singular zero with a given multiplicity structure up to a given order.…”

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

“…In recent years, Mourrain et al [10,11] propose a new deflation method, which can be used to refine the accuracy of an isolated singular zero and the parameters introduced simultaneously and what's more, the parameters can describe the multiplicity structure at the zero. They also prove that the number of equations and variables in this deflation method depends polynomially on the number of variables and equations of the input system and the multiplicity of the singular zero.…”

“…Theorem 16. [11] Let K ⊂ C be any field, F ⊂ K[x], and let p ∈ C n be an isolated zero of F. Let Q be the primary ideal at p and assume that B is a basis for K[x]/Q satisfying the conditions of Lemma 13. Let E ⊂ N n be as in Lemma 13 and M i (µ) for i = 1, .…”