JournalscmhVol. 74 , No. 3DOI 10.1007/s000140050094

The Łojasiewicz exponent of an analytic function at an isolated zero

  • Janusz Gwoździewicz

    Pedagogical University of Cracow, Kraków, Poland
The Łojasiewicz exponent of an analytic function at an isolated zero cover

Abstract

Let f be a real analytic function defined in a neighborhood of 0Rn0 \in {\Bbb R}^n such that f1(0)={0}f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: f(x)cxα|f(x)|\geq c|x|^{\alpha} , gradf(x)cxβ|{\rm grad}\,f(x)|\geq c|x|^{\beta} , gradf(x)cf(x)θ|{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c>0c > 0 . We prove that α=β+1\alpha=\beta+1, θ=β/α\theta=\beta/\alpha . Moreover β=N+a/b\beta=N+a/b where 0ha<bhNn10 h a < b h N^{n-1} . If f is a polynomial then f(x)cx(degf1)n+1|f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.