Global Łojasiewicz-type inequality for non-degenerate polynomial maps

Let F : = ( f 1 , … , f p ) : R n → R p be a polynomial map. This paper studies the existence of the following global Łojasiewicz-type inequality ‖ F ( x ) ‖ α + ‖ F ( x ) ‖ β ⩾ c d ( x , F − 1 ( 0 ) ) for all x ∈ R n , for some constants c > 0 , α > 0 , and β > 0 . We show that the above inequality holds if one of the following conditions is satisfied:(i) onvenient and Khovanskii non-degenerate at infinity; onvenient and non-degenerate at infinity; ikhailov–Gindikin non-degenerate. er, in cases (ii) and (iii), the exponents α and β can be determined explicitly.