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On perturbation of roots of homogeneous algebraic systems
Authors:S Tanabé    M N Vrahatis
Institution:Department of Mathematics, Independent University of Moscow, Bol'shoj Vlasievskij pereulok 11, 121002 Moscow, Russia ; Computational Intelligence Laboratory (CI Lab), Department of Mathematics, University of Patras Artificial Intelligence Research Center (UPAIRC), University of Patras, GR--26110 Patras, Greece
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.

Keywords:Polynomial systems  location of zeros
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