Growth at infinity and index of polynomial maps
Let \(F:\mathbb{K}^{n}\mathbb{\rightarrow K}^{n}\) be a polynomial mapping such that \(F^{-1}(0)\) is compact, where \(\mathbb{K=R}\) or \(\mathbb{C}\). The authors give an estimation of the Łojasiewicz exponent at infinity of \(F\) (defined as the supremum of those real numbers \(\alpha \) for which there exist constants \(C,M>0\) such that \(\left| \left| x\right| \right| ^{\alpha }\leq C\left| \left| F(x)\right| \right| \) for \(\left| \left| x\right| \right| \geq M),\) under some non-degeneracy conditions for some Newton polyhedra. This estimation allows a uniform estimation of the Łojasiewicz exponent at infinity for the mappings in a homotopy \(F+tG,\) \(t\in [ 0,1]\), where \(G \) is another polynomial mapping satisfying appropriate conditions. This, in turn, implies the invariance of the global index of the mappings in the homotopy \(F+tG.\) In particular, \(F\) and \(F+G\) have the same global index.
scientific article; zbMATH DE number 6362884
scientific article