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28 March 2024
 
  » arxiv » 1910.5203

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Multiplicity and degree relative to a set
Vincent Grandjean ; Maria Michalska ;
Date 11 Oct 2019
AbstractThe 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 ${Gamma_d}_{din mathbb{N}}$, 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 $(S_t)_{tinmathbb{R}^m}$ of sets given by inequalities $f_i+t_ig_igeq 0$ 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, 1910.5203
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