On perturbation of roots of homogeneous algebraic systems

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.