On the global Łojasiewicz inequality for polynomial functions
Volume 122 / 2019
Abstract
Let $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ be a polynomial in $n$ variables. We study the following global Łojasiewicz inequality for $f$: \[ |f(x)|\geq c\min \{{\rm dist}(x,f^{-1}(0))^\alpha, {\rm dist}(x,f^{-1}(0))^\beta\} \] for all $x\in\mathbb{R}^n,$ where $c, \alpha, \beta$ are positive constants. Let $f$ be written in the form $$ f(x_1,\ldots,x_n)=a_0x_n^{d}+a_1(x’)x_n^{d-1}+ \cdots +a_d(x’), $$ where $d$ is the degree of $f$ and $x’=(x_1,\ldots, x_{n-1}).$ We prove that the global Łojasiewicz inequality for $f$ holds for all $x\in\mathbb{R}^n$ if and only if it holds for all $x\in V_1:= \{x\in\mathbb{R}^n : \partial f/\partial x_n=0 \}.$ For $n=2$, this gives a simple method for checking the existence of the global Łojasiewicz inequality. We will consider the following problems for $n=2$: (a) computation of global Łojasiewicz exponents; (b) studying the global Łojasiewicz inequality for polynomials which are non-degenerate at infinity; (c) computation of the exponent involved in the Hörmander version of the global Łojasiewicz inequality.
