Łojasiewicz inequality on non-compact domains and singularities at infinity (Q2863009)

The main result of the paper is a version of the Łojasiewicz inequality for real polynomials on non-compact domain: for a polynomial \(f:\mathbb{R}^n\to\mathbb{R}\), \(n\geq 2\), monic of positive degree \(m\) in the variable \(x_1\) the following Łojasiewicz inequality holds NEWLINE\[NEWLINE |f(x)|\geq c \text{dist} (x,\tilde V)^\alpha \text{ for some } c,\,\alpha>0,\text{ as }f(x)\to 0,\tag{1} NEWLINE\]NEWLINE where \(\tilde V\) is a union of the zero-set \(V\) of \(f\) and the polar set \(V_1=(\partial f/\partial x_1)^{-1}(0)\). It improves the result of \textit{D. S. Tiep, H. H. Vui} and \textit{N. T. Thao} [Int. J. Math. 23, No. 4, Article ID 1250033, 28 p. (2012; Zbl 1246.26013)], where as \(\tilde V\) is taken a union of \(V\) and \(V_i=({\partial^i f}/{\partial x^i_1})^{-1}(0)\), \(i=1,\ldots,m-1\).NEWLINENEWLINEThe inequality (1) is used in the proof of the next result of this paper which is a sufficient condition for the existence of vanishing component at infinity for a polynomial \(f\) as \(t\to 0\) (Theorem 3.1). Recall that \(f\) has a vanishing component as \(t\to 0\) if for every \(\varepsilon,R>0\) there exists \(t\in(-\varepsilon,+\varepsilon)\) such that \(f^{-1}(t)\) admits a connected component in the set \(\mathbb{R}^n\setminus\mathbb{B}_R\), where \(\mathbb{B}_R=\{x\in\mathbb{R}^n:|x|\leq R\}\). Theorem 3.1 says that if \(f\) has a sequence of first type (i.e. a sequence \(x^k\to \infty\) in \(\mathbb{R}^n\) such that \(f(x^k)\to0\) and \(\text{dist} (x^k,V)\geq \delta\) for some \(\delta>0\)) and distance of \(V\) and \(V_1\setminus \mathbb{B}_r\) is positive as \(r\to \infty\), then \(f\) admits a vanishing component at infimnity as \(t\to0\). In particular \(0\) is an atypical value of \(f\). Example 3.1 shows that existence of a sequence of the first type does not necessarily imply existence a vanishing component at infinity.