Detection methods of asymptotic critical values of polynomial mappings

Abstract

In algebraic geometry, the problem of detecting the bifurcation values of a polynomial is very important. The bifurcation values of a polynomial mapping consist of the bifurcation value at infinity and the set of critical values of its. This problem is generally encountered as detecting bifurcation value at infinity, which is a subset of the bifurcation values of the polynomial. This corresponds to determining some supersets containing bifurcation values at infinity. In addition, it is another important problem to determine the cases where the bifurcation values consist only of the values of the polynomial at the critical points. This is equivalent to bifurcation values at infinity is empty. In this thesis, we firstly construct a curve that approaching an asymptotic critical value which is a superset of the bifurcation value at infinity with very few coefficients. We used toric geometry as the main tool. By aids of, we get the corollary that says every critical value of polynomial mappings over the bad face of Newton polyhedron is an element of asymptotic critical value. Finally, we give a method to construct a curve approaching an asymptotic critical value of a real polynomial map, corresponding to detect real coefficients of the parametric representation of the curve. Asymptotic critical values sometimes correspond to the infimum or supremum of the polynomial. We hope that the study can be applied to optimization problems.