A global smooth version of the classical Łojasiewicz inequality

Volume estimates of sublevel sets of real polynomials

“…We begin with a global Lojasiewicz inequality which is of interest in its own right. This result is analogous to Theorem 2.1 in [11] where only smoothness of functions is taken into account. The following notions related to a polynomial P ∈ R[x 1 , · · · , x n ] are crucial in our work.…”

“…Theorem 3.1). We should say that similar (but weaker) global Lojasiewicz inequalities were obtained earlier (see [4,Theorem 4.6]) and [11,Theorem 2.1]). One novelty in our work is to provide a sharper Lojasiewicz inequality where the notion of admissible monomial is taken into account (cf.…”

“…Proof. We use an idea in the proof of the main theorem in [5] (see also Theorem 2.1 in [11]). After a permutation of coordinates, we achieve that i j = j for every 1 ≤ j ≤ n. So we can write…”

“…We should say that in the case k = 2, this property of "separated at infinity" is essentially equivalent to that of "non-asymptotic at infinity" introduced in [11]. For simple examples of semi-algebraic sets which are separated at infinity we may take a finite number of real lines in R 2 that pass through the origin.…”

The Łojasiewicz Exponent at Infinity of Non-negative and Non-degenerate Polynomials

“…for some positive constants δ, C, α, where Z = {x ∈ R n : f (x) • ∂ f ∂ x 1 (x) = 0}, and f is a monic polynomial with respect to x 1 . In this case constants C, α can be computed explicitly (see Hà et al 2015).…”