Global Łojasiewicz-type inequality for non-degenerate polynomial maps

Let F:=(f1,…,fp):Rn→RpF:=(f1,…,fp):Rn→Rp be a polynomial map. This paper studies the existence of the following global Łojasiewicz-type inequality‖F(x)‖α+‖F(x)‖β⩾cd(x,F−1(0))for allx∈Rn, for some constants c>0,α>0c>0,α>0, and β>0β>0. We show that the above inequality holds if one of the following conditions is satisfied:(i)F is convenient and Khovanskii non-degenerate at infinity;(ii)F is convenient and non-degenerate at infinity;(iii)F is Mikhailov–Gindikin non-degenerate. Further, in cases (ii) and (iii), the exponents α and β can be determined explicitly.