An explicit bound for the Łojasiewicz exponent of real polynomials (Q447784)

Let \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be an analytic function defined in a neighbourhood \(U\) of \(0\in \mathbb{R}^{n}\), \(f(0)=0\), and \(Z=\{x\in U:f(x)=0\}\). The Łojasiewicz exponent \(\alpha _{f}\) of \(f\) at \(0\) is the infimum of the exponents \(\alpha \) for which there exist constants \(r>0\), \( c>0 \) such that the Łojasiewicz inequality NEWLINE\[NEWLINE|f(x)| \geq c\,\mathrm{dist}(x,Z)^{\alpha }NEWLINE\]NEWLINE holds for \(\| x\| \leq r\). The author proves the estimation NEWLINE\[NEWLINE\alpha_f\leq \max \big\{d(3d-4)^{n-1},2d(3d-3)^{n-2}\big\} NEWLINE\]NEWLINE of \(\alpha_f\) for any polynomial \(f\in \mathbb{R}[x_1,\dots,x_n]\) of degree \(d\). The same result has been obtained (in a different way) by \textit{K. Kurdyka} and \textit{S. Spodzieja} [Proc. Am. Math. Soc., in press]. It generalizes a result by \textit{J. Gwoździewicz} [Comment. Math. Helv. 74, No. 3, 364--375 (1999; Zbl 0948.32028)] in the case \(f\) has an isolated zero at \(0\) .