Effective Lojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs
The main result is an effective estimation of the best exponent \(\rho \in \lbrack 0,1)\) in the Lojasiewicz gradient inequality \[ \left\vert \nabla f(x)\right\vert \geq C\left\vert f(x)\right\vert ^{\rho } \] in a neighbourhood of \(a\in U\subset \mathbb{R}^{n}\) for some constant \(C>0,\) for a Nash function \(f:U\rightarrow \mathbb{R},\) where \(f(a)=0.\) Namely, if \(d\) is the degree of a nonzero polynomial \(P\) such that \( P(x,f(x))=0,\) \(x\in U,\) then the above inequality holds for \[ \rho =1-\frac{1}{2(2d-1)^{3n+1}}. \] This is a generalization of the \textit{P. Solernó} [Appl. Algebra Eng. Commun. Comput. 2, No. 1, 1--14 (1991; Zbl 0754.14035)] and \textit{D. D'Acunto} and \textit{K. Kurdyka} [Ann. Pol. Math. 87, 51--61 (2005; Zbl 1093.32011)] type estimation for polynomials. As a corollary the authors obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.
scientific article; zbMATH DE number 7317371
scientific article