  The Lojasiewicz inequalities for real analytic functions on Euclidean space
were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic
and subanalytic sets, arguments later simplified by Bierstone and Milman
(1988). In this article, we first give an elementary geometric,
coordinate-based proof of the Lojasiewicz 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 analytic varieties, that the gradient inequality for an arbitrary real or
complex analytic function follows from the special case where it has simple
normal crossings. In addition, we prove the Lojasiewicz inequalities when a
function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an
elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and
Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349
(infinite-dimensional case).
