论文标题
NASH功能的Bernstein-Remez不等式：一种复杂的分析方法
Bernstein-Remez inequality for Nash functions: A complex analytic approach
论文作者
论文摘要
考虑一个开放的，有限的集合$ω\ subset \ mathbb {c} $，一个正整数$ k $和一个紧凑型$ \ mathcal {k} \ subsetω$严格大于$ k $。我们证明，对于任何功能$ f $，这是$ \ operlineω$中的holomorphic，并且其图表满足$ s（z，f（z））= 0 $，对于某些多项式$ s \ in \ mathbb {c} [c} [z，w] $ k $（因此，$ f $ f $ as a algebraic formatity in \ mathbb {c} [z，w] $ $ \ max _ {\overlineΩ} | f |/\ max _ {\ mathcal {k}} | f | $受一个不取决于$ k $，$ω$，$ \ nathcal {k} $的常数的界限，但不依赖于$ f $（$ f $ f $ f $ for $ f $）。 Roytwarf和Yomdin已证明了这一结果，如果$ \ Mathcal {K} $是一个真正的间隔，而Yomdin则使用真实的和分析几何的参数，用于离散的$ \ MATHCAL $ \ MATHCAL {k} $。在这里，由于Nekhoroshev对代数函数存在统一的伯恩斯坦雷姆斯不平等的不平等现象，我们介绍并扩展了证明，这依赖于复杂分析的经典定理。 Nekhoroshev的作品仍然没有研究，尽管它在哈密顿动力学中产生了重要影响，并且在这里以独立和教学的方式进行了介绍和扩展，而原始的理由则相当粗略。
Consider an open, bounded set $Ω\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset Ω$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline Ω$, and whose graph satisfies $S(z,f(z))=0$ for some polynomial $S\in\mathbb{C}[z,w]$ of degree at most $k$ (hence $f$ is an algebraic function), the quantity $\max_{\overlineΩ}|f|/\max_{\mathcal{K}}|f|$ is bounded by a constant that only depends on $k$, $Ω$, $\mathcal{K}$ but not on $f$ (estimates of this kind are called Bernstein-Remez inequalities). This result has been demonstrated by Roytwarf and Yomdin in case $\mathcal{K}$ is a real interval, and later by Yomdin for a discrete set $\mathcal{K}$ of sufficiently high cardinality, by using arguments of real-algebraic and analytic geometry. Here we present and extend a proof due to Nekhoroshev on the existence of a uniform Bernstein-Remez inequality for algebraic functions, which relies on classical theorems of complex analysis. Nekhoroshev's work remained unstudied despite its important consequences in Hamiltonian dynamics and is here presented and extended in a self-contained and pedagogical way, while the original reasonings were rather sketchy.
