A+ CATEGORY SCIENTIFIC UNIT

Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Annales Polonici Mathematici

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On the global Łojasiewicz inequality for polynomial functions

Volume 122 / 2019

Annales Polonici Mathematici 122 (2019), 21-47 MSC: 14P05, 14P15, 14H50. DOI: 10.4064/ap171126-21-9 Published online: 15 February 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.

Authors

• Huy Vui HaThang Long Institute of Mathematics
and Applied Sciences
Nghiem Xuan Yem Road, Hoang Mai District
Ha Noi, Vietnam
e-mail
• Van Doat DangThang Long High School for the Gifted