  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.
