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On the Łojasiewicz Exponent near the Fibre of a Polynomial

100%

Skalski G.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

tom 52

nr 3

231-236

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

Sum of squares and the Łojasiewicz exponent at infinity

51%

Kurdyka K., Osińska-Ulrych B., Skalski G., Spodzieja S.

Annales Polonici Mathematici

2014

tom 112

nr 3

223-237

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Bulletin of the Polish Academy of Sciences. Mathematics

2 Skalski G.

1 Kurdyka K.

1 Osińska-Ulrych B.

1 Spodzieja S.

1 2014

1 2004

Wyniki wyszukiwania

On the Łojasiewicz Exponent near the Fibre of a Polynomial

Sum of squares and the Łojasiewicz exponent at infinity