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
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
and generalized
Morse–Bott of order
;
we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is
and
Morse–Bott on a Banach space.
Keywords
analytic varieties, Łojasiewicz inequalities, gradient
flow, Morse–Bott functions, resolution of singularities,
semianalytic sets and subanalytic sets
