The Łojasiewicz exponent of weighted homogeneous polynomials at infinity

Abstract

Let $f:{\mathbb{C}}^n⟶\mathbb{C}$ be a weighted homogeneous polynomial with a finite number of critical points. The aim of this paper is to give a formula for the Łojasiewicz exponent of f at infinity in terms of its weights $w_i$, when $w_i\ge 2$. We also give a characterization of properness for weighted homogeneous polynomial mapppings. As a corollary, we prove the Jacobian conjecture for weighted and preweighted homogeneous polynomial mappings with positive weights.