Let \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a polynomial function of degree \(d\) with \(f(0)=0\) and \(\nabla f(0)=0\) (\(\nabla f:=\text{grad}f).\)
The authors give an explicit bound for the best exponent\(\rho _{f}:=\inf (\rho )\) in the Łojasiewicz inequality \[ \left| \nabla f(x)\right| \geq C\left| f(x)\right| ^{\rho }, \] where \(C\) is a positive constant and \(x\) is in a sufficiently small neighbourhood of \(0.\) Namely \[ \rho _{f}\leq 1-\frac{1}{d(3d-3)^{n-1}}. \] The same result has been obtained by A. Gabrielov in [Sel. Math., New Ser. 1, 113–127 (1995; Zbl 0889.32005)]. This result generalizes the result by J. Gwoździewicz in the isolated case (i.e. \(\nabla f\) has an isolated zero at \(0\)) [Comment. Math. Helv. 74, No. 3, 364–375 (1999; Zbl 0948.32028)].