Sum of squares and the Łojasiewicz exponent at infinity

• Autorzy: Krzysztof Kurdyka , Beata Osińska-Ulrych , Grzegorz Skalski , Stanisław Spodzieja
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 14R99: None of the above, but in this section
• 14P05: Real algebraic sets
• 32S70: Other operations on singularities
• 11E25: Sums of squares and representations by other particular quadratic forms

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2014
• Tom: 112
• Numer: 3
• Strony: 223-237

Daty

wydano

2014

Twórcy

autor

Krzysztof Kurdyka

• Laboratoire de Mathématiques (LAMA), Université de Savoie, UMR-5127 de CNRS, 73-376 Le Bourget-du-Lac Cedex, France

autor

Beata Osińska-Ulrych

• Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Łódź, Poland

autor

Grzegorz Skalski

• Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Łódź, Poland

autor

Stanisław Spodzieja

• Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Łódź, Poland

Identyfikatory

DOI

10.4064/ap112-3-2