Özet:
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.
