On perturbation of roots of homogeneous algebraic systems

Abstract

Abstract. A problem concerning the perturbation of roots of a system of homogeneous algebraic equations is investigated. The question of conservation and decomposition of a multiple root into simple roots are discussed. The main theorem on the conservation of the number of roots of a deformed (not necessarily homogeneous) algebraic system is proved by making use of a homotopy connecting initial roots of the given system and roots of a perturbed system. Hereby we give an estimate on the size of perturbation that does not affect the number of roots. Further on we state the existence of a slightly deformed system that has the same number of real zeros as the original system in taking the multiplicities into account. We give also a result about the decomposition of multiple real roots into simple real roots.

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

“…at x * it holds that det J Fn (x * ) = 0, otherwise it is called multiple. The problem of conservation and decomposition of a multiple root into simple roots in the case of systems of homogeneous algebraic equations has been tackled in [57]. This approach can be applied to high dimensional CAD where it is sometimes required to compute the intersection of several hypersurfaces that are a perturbation of a set of original unperturbed hypersurfaces.…”

Survey of Methods for Solving Systems of Nonlinear Equations, Part II: Optimization Based Approaches

“…at x * it holds that det J Fn (x * ) = 0, otherwise it is called multiple. The problem of conservation and decomposition of a multiple root into simple roots in the case of systems of homogeneous algebraic equations has been tackled in [57]. This approach can be applied to high dimensional CAD where it is sometimes required to compute the intersection of several hypersurfaces that are a perturbation of a set of original unperturbed hypersurfaces.…”

Adjacency spectra of random and complete hypergraphs

“…The step of showing that the derivative is bounded by a norm-like function of A − B is the next main problem to address in this program, and is made difficult both by the fact that matrix-theoretic proofs of analogous results use smoothness of the eigenvectors (something that is difficult to generalize to hypermatrices because the singular surfaces involved are no longer linear spaces but some more complicated projective varieties) and ordinary norms are not in general holomorphic (because they involve the modulus function). However, there is reason to believe that the relatively simple differential behavior of algebraic sets along curves will permit the effective use of homotopy techniques (tools such as [42,39,33]).…”