Year: 2019
Type: preprint
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 $\{Γ_d\}_{d\in \mathbb{N}}$, 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 $Γ_d$. 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.
Source: arXiv (Cornell University)
Authors: Grandjean, Vincent, Michalska, Maria
Institutions: Universidade Federal do Ceará, University of Łódź
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3041655236, G4402799499
OpenAlex ID: W2979392809