Multiplicity and degree relative to a set

TL;DR

This paper introduces a method to compute the relative multiplicity and degree of functions with respect to semianalytic sets using a family of curves, and establishes stratification results for parameterized families of sets.

Contribution

It constructs a family of curves determining the relative multiplicity of polynomials and proves stratification of parameter spaces where this multiplicity remains constant.

Findings

Existence of a curve family $\u201c\Gamma_d$ for relative multiplicity calculation.

Stratification of parameter space for semianalytic families where multiplicity is invariant.

Analogous algebraic results for relative degree under algebraic data.

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 $|f(x)|\leq C\|x\|^j$ 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 $f_i+t_ig_i\geq 0$ 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.