Stanisław Spodzieja

Separation of real algebraic sets and the Łojasiewicz exponent

We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz’s exponents in the real and complex setting.

Metric Properties of Semialgebraic Mappings

We give various quantitative versions of Łojasiewicz inequalities for semialgebraic sets and mappings, both in the local and global case.

LOJASIEWICZ EXPONENT NEAR THE FIBRE OF A MAPPING

Let g: X → ℝ k and f: X → ℝ m , where X C ℝ n , be continuous semi-algebraic mappings, and λ ∈ ℝ m . We describe the optimal exponent θ =: L ∞,f→λ (g) for which the Lojasiewicz inequality |g(x)| ≥ C|x| θ holds with C > 0 as |x| → ∞ and f(x) → λ. We prove that there exists a semi-algebraic stratification ℝ m = S 1 ∪ ... U S j such that the function λ ↦ L ∞,f→λ (g) is constant on each stratum S i . We apply this result to describe the set of generalized critical values of f.

Convexifying Positive Polynomials and Sums of Squares Approximation

We show that if a polynomial

The Łojasiewicz exponent of subanalytic sets

f\in \mathbb{R}[x_1,\ldots,x_n]

Multiplicity and the Łojasiewicz exponent

is nonnegative on a closed basic semialgebraic set

Effective formulas for the local Łojasiewicz exponent

X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}

Effective formulas for the Łojasiewicz exponent at infinity

, where

The Łojasiewicz gradient inequality in a neighbourhood of the fibre

g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]

Łojasiewicz inequalities at infinity for the gradient of a polynomial

, then