Finding Isolated Zero Points of Complex Polynomial Mappings.

摘要

In this paper, at first the properties of isolated zero points of complex polynomial mappings are discussed using the integral expression of a degree of C(sup 1) mapping, Sard's Theorems and homotopy. Then we prove that for almost every complex polynomial mapping P:C(sup n)->C(sup n), the zero set H(sup -1)(0) of the homotopy H(z,t)=tP(z)+(1-t)Q(z) consists of s=PI(sub j=1)(sup n)s(sub j) disjoint differential curves, and the zero set phi(sub (delta)(sub j))(sup -1)(0) of its piecewise linear approximation phi(sub (delta)(sub j)) consists of some broken lines which do not meet (nu)-dimensional simplexes (0(le)(nu)(le)2n-1), where Q(z)=(Q(sub 1)(z),...,Q(sub n)(z)), Q(sub j)(z)=z(sub j)(sup sj)-b(sub j)(sup sj), b(sub j)not =0,j=1,...,n. When (lim(sub j->)(infinity)(delta)(sub j)=0), these broken lines tend to corresponding differential curves in H(sup -1)(0). (author). 4 refs. (Atomindex citation 20:046718)