Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Abstract

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). Here we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for (real or complex) analytic varieties, that the gradient inequality for an arbitrary analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Łojasiewicz inequalities when a function is $C^N$ and generalized Morse–Bott of order $N\ge 3$; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is $C^2$ and Morse–Bott on a Banach space.