Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, et al. "Sum of squares and the Łojasiewicz exponent at infinity." Annales Polonici Mathematici 112.3 (2014): 223-237. <http://eudml.org/doc/286262>.

@article{KrzysztofKurdyka2014,
abstract = {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.},
author = {Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja},
journal = {Annales Polonici Mathematici},
keywords = {polynomial mapping; extension; Łojasiewicz exponent at infinity; sum of squares; Positivstellensatz},
language = {eng},
number = {3},
pages = {223-237},
title = {Sum of squares and the Łojasiewicz exponent at infinity},
url = {http://eudml.org/doc/286262},
volume = {112},
year = {2014},
}

TY - JOUR
AU - Krzysztof Kurdyka
AU - Beata Osińska-Ulrych
AU - Grzegorz Skalski
AU - Stanisław Spodzieja
TI - Sum of squares and the Łojasiewicz exponent at infinity
JO - Annales Polonici Mathematici
PY - 2014
VL - 112
IS - 3
SP - 223
EP - 237
AB - 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.
LA - eng
KW - polynomial mapping; extension; Łojasiewicz exponent at infinity; sum of squares; Positivstellensatz
UR - http://eudml.org/doc/286262
ER -