Łojasiewicz inequality at infinity for a pair of polynomials and some applications

Let $P,Q: \mathbb R^n \rightarrow \mathbb R$ be two polynomials. This paper studies the existence of the following Łojasiewicz inequality at infinity: $$|Q(x)|^{\theta} \ge c|P(x)| \quad\ {\rm for}\ \|x\| \gg 1,$$ where $c$ and $\theta $ are positive constants. We provide a condition under which the Łojasiewicz inequality holds, and the exponent is computed explicitly in terms of the Newton polyhedra of the two polynomials. On the way, we give some criteria for the convergence of some integrals of rational functions, and describe the domain of convergence of multidimensional Dirichlet series associated with polynomials in terms of Newton polyhedra of polynomials defining the series.