论文标题
改善了有效的a ojasiewicz不平等和应用
Improved effective Łojasiewicz inequality and applications
论文作者
论文摘要
令$ \ mathrm {r} $为一个真正的封闭字段。给定一个封闭且有限的半代数集$ a \ subset \ subset \ mathrm {r}^n $和半代数连续函数$ f，g：a \ rightarrow \ mathrm {r} $，使得$ f^{ - 1}（-1}（0） \ mathrm {r} $，使得不等式（lojasiewicz不等式）$ | g（x）|^n \ le c \ cdot | f（x）| $ co $ hold in $ x \ in a $ in a $。在本文中，我们考虑了$ a $由$ p = 0，p> 0的原子定义的$ a $定义的情况。 $ f，g $的图也由无量化式公式定义，其原子的原子为$ q = 0，q> 0，q \ in \ mathcal {q} $，对于某些有限的集合$ \ mathcal {q} \ subset {q} \ subset \ subset \ subset \ mathrm {r} [x_1，_1，x_1，x_1，\ ldots，x________________________________________________________________________________]我们证明，在这种情况下，lojasiewicz指数$ n $由$（8 d）^{2（n+7）} $界定。我们的界限取决于$ d $和$ n $，但独立于组合参数，即$ \ Mathcal {p} $和$ \ Mathcal {q} $的基础性。结果，我们在某些条件下改善了多项式系统的当前最佳误差范围。最后，作为从给定函数的描述中出现的组合参数的独立性概念的抽象，我们证明了在多个界限的O-Minimal结构中的lojasiewicz不平等的版本。我们证明了某些合并定义的无限（但不一定是可定义的）函数对的共同oljasiewicz指数的存在。
Let $\mathrm{R}$ be a real closed field. Given a closed and bounded semi-algebraic set $A \subset \mathrm{R}^n$ and semi-algebraic continuous functions $f,g:A \rightarrow \mathrm{R}$, such that $f^{-1}(0) \subset g^{-1}(0)$, there exist $N$ and $c \in \mathrm{R}$, such that the inequality (Łojasiewicz inequality) $|g(x)|^N \le c \cdot |f(x)|$ holds for all $x \in A$. In this paper we consider the case when $A$ is defined by a quantifier-free formula with atoms of the form $P = 0, P >0, P \in \mathcal{P}$ for some finite subset of polynomials $\mathcal{P} \subset \mathrm{R}[X_1,\ldots,X_n]_{\leq d}$, and the graphs of $f,g$ are also defined by quantifier-free formulas with atoms of the form $Q = 0, Q >0, Q \in \mathcal{Q}$, for some finite set $\mathcal{Q} \subset \mathrm{R}[X_1,\ldots,X_n,Y]_{\leq d}$. We prove that the Łojasiewicz exponent $N$ in this case is bounded by $(8 d)^{2(n+7)}$. Our bound depends on $d$ and $n$, but is independent of the combinatorial parameters, namely the cardinalities of $\mathcal{P}$ and $\mathcal{Q}$. As a consequence we improve the current best error bounds for polynomial systems under some conditions. Finally, as an abstraction of the notion of independence of the Łojasiewicz exponent from the combinatorial parameters occurring in the descriptions of the given pair of functions, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions.
