Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka ; Beata Osińska-Ulrych ; Grzegorz Skalski ; Stanisław Spodzieja

Annales Polonici Mathematici, Tome 111 (2014), p. 223-237 / Harvested from The Polish Digital Mathematics Library

Résumé

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁(x) = ⋯ = h_{r}(x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then $f|_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h(x) = ∑_{i=1}^{r} h²_{i}(x)σ_{i}(x)$, where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x ∈ ℝⁿ. We give an estimate for p in terms of: the degree of f, the degrees of $h_{i}$ and the Łojasiewicz exponent at infinity of $f|_{V}$. We prove a version of the above result for polynomials positive on semialgebraic sets. We also obtain a nonnegative extension of some odd power of f which is nonnegative on an irreducible algebraic set.

Détail

Publié le : 2014-01-01

Zbl 1315.14073

EUDML-ID : urn:eudml:doc:286262