Metric Properties of Semialgebraic Mappings

Abstract

We give an effective estimation from above for the local Łojasiewicz exponent for separation of semialgebraic sets and for a semialgebraic mapping on a closed semialgebraic set. We also give an effective estimation from below of the Łojasiewicz exponent in the global separation for semialgebraic sets and estimation of the Łojasiewicz exponent at infinity of a semialgebraic mapping similar to the Jelonek result [14] in the complex case. Moreover, we prove that both local and global Łojasiewicz exponent of an overdetermined semialgebraic mapping F : X → R m on a closed semialgebraic set X ⊂ R n (i.e. m > dim X) are equal to the Łojasiewicz exponent of the composition L • F : X → R n for the generic linear mapping L : R m → R k , where k = dim X.

Some estimations of the Lojasiewicz exponent for polynomial mappings on semialgebraic sets

“…with R = r(X) + r(graph F ). Actually in [5] there is no m in the inequality but this should be considered a typographical error. Thus in our theorem we do improve their estimation by using r = r(X) instead of R = r(X) + r(graph F ).…”

“…Thus in our theorem we do improve their estimation by using r = r(X) instead of R = r(X) + r(graph F ). For this paper, to be self-contained and more clear, we will have to repeat some of the argumentation from [5] for polynomial mappings on semialgebraic sets. In the proof of Theorem 1 we will use the result obtained in [4, Corollary 8] regarding Lojasiewicz exponent in the case of two algebraic sets.…”

“…Using this, in [5,Corollary 3.4] it has been shown that for a polynomial mapping F on a closed semialgebraic set X the following inequality holds:…”

“…Proof of Theorem 2. Again we shall repeat the argumentation from [5]. Also, as in the previous proof we will consider the set X × {0} ⊂ R N × R m defined by (3), and denote it simply by X to avoid overuse of notation.…”

“…Estimates of the Lojasiewicz exponent are nowadays widely used in real and complex algebraic geometry. Kudryka, Spodzieja and Szlachcińska in [5] have given an estimate of the Lojasiewicz exponent at a point for a continuous semialgebraic mapping on a closed semialgebraic set and an estimate of the Lojasiewicz exponent at infinity for a polynomial mapping on a semialgebraic set. In this paper we show that in case of a polynomial mapping, at a point or at infinity, it is possible to obtain slightly stronger results than they have.…”