Bernstein-Remez inequality for Nash functions: A complex analytic approach

Consider an open, bounded set $\Omega\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset \Omega$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline \Omega$, 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 $\max_{\overline\Omega}|f|/\max_{\mathcal{K}}|f|$ is bounded by a constant that only depends on $k$, $\Omega$, $\mathcal{K}$ 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.