Multiplicity and degree relative to a set

Abstract

The multiplicity (resp. degree) of a function $f$ relative to a semianalytic subset $S$ of ${\mathbb{R}}^n$ is the greatest (resp. smallest) exponent among numbers $j$ such that the inequality $|f(x)|\le C\Vert x{\Vert }^j$ holds on $S$ near $0$ (resp. near infinity) for some constant $C$. We show that there exists a family of curves $\{{\Gamma }_d{\}}_{d\in \mathbb{N}}$, determined only by the set, such that the relative multiplicity of any polynomial of degree $d$ is equal to its relative multiplicity with respect to ${\Gamma }_d$. Moreover, a semianalytic family $(S_t{)}_{t\in {\mathbb{R}}^m}$ of sets given by inequalities $f_i+t_i{g}_i\ge 0$ for $i=1,\dots ,m$ admits a stratification of the parameter space ${\mathbb{R}}^m$ such that on each component of the top-dimensional stratum the relative multiplicity function on ${\mathcal{O}}_n$ does not change. Analogous results, assuming the data are algebraic, hold in the relative degree case.

30 pages