# Effective {\L}ojasiewicz gradient inequality and finite determinacy of non-isolated Nash function singularities

## TL;DR

This paper establishes bounds on the Łojasiewicz gradient inequality exponent for Nash functions on semialgebraic sets, linking it to polynomial degrees, and explores finite determinacy of non-isolated Nash singularities.

## Contribution

It provides a new estimation method for the Łojasiewicz exponent based on polynomial degrees and applies this to analyze finite determinacy of Nash singularities.

## Key findings

- Derived bounds for the Łojasiewicz gradient inequality exponent.
- Linked the exponent estimation to the degree of a defining polynomial.
- Provided criteria for finite determinacy of non-isolated Nash singularities.

## Abstract

Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the {\L}ojasiewicz gradient inequality $|\nabla f(x)|\ge C|f(x)|^\varrho$ for $x\in X$, $|f(x)|<\varepsilon$ for some constants $C,\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities