On the Number of Homotopy-Invariant Solutions for Systems of Polynomial Equations

Abstract

There are many applications which lead to systems of polynomial equations (e.g. the application of the Kuhn-Tucker conditions to suitable nonlinear optimization problems). One way of solving systems of polynomial equations is by homotopy algorithms (deformation algorithms). One can set up homotopies in ℝⁿ, but then one has to show (i) that homotopy-paths do not cross, and (ii) that the homotopy-paths converge to the right solutions. One can impose stricter requirements and ask the following question: Is it possible to set up homotopies in a suitable space in such a way as to ensure that the number of solutions one starts with is a homotopy-invariant in this space. Homotopies in ℝⁿ in general do not satisfy this strict requirement (because ℝⁿ can be deformed into a point, or all homotopy-paths may converge to the same solution).