An alternative proof of the Łojasiewicz inequality

Question

Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., standard Calculus 1+2). I admit i hadn't time to think about it properly, I thought I could prove it first for homogeneous polynomials (and I got stuck) and then extend the result to analytic functions, but I could be wrong... To avoid misunderstandings I recall here the statement of the Łojasiewicz inequality.

Let $f\in C^{\omega}(B_{1},\mathbb{R})$ (real analytic function on the open unit ball centered at the origin) then there are $\alpha\in (0,1)$, $C>0$ and $\varepsilon$ sufficiently small such that $\forall x\in B_{\varepsilon}$ the following holds true: $$|f(x)-f(0)|^{\alpha}\leq C|\nabla f(x)|.$$

Thank you in advance.

Answer 1

Rate of decay to equilibrium in some semilinear parabolic equations, Alain Haraux, Mohamed Ali Jendoubi, Otared Kavian (2003).

I would think section 2, "A direct proof of the Lojasiewicz inequality" has what you're looking for:

In this paper we are going to present an elementary proof of the Lojasiewicz inequality in the framework of an energy functional, under simple circumstances but without assuming the analyticity of the nonlinearity of the function $f$.

An alternative (somewhat less "direct") proof is given in On the Łojasiewicz–Simon gradient inequality, Ralph Chill (2003).

Answer 2

No, there is no elementary proof of that result even for polynomials of 2 variables. The proof of this result an be reduced to a more elementary one (the case of monomials) by using the theory of sigularities, but this is not at all elementary, cf. the recent works of Paul Feehan.