Bernstein–Remez inequality for algebraic functions: A topological approach

Abstract

By taking full advantage of the structure of complex algebraic curves and by using compactness arguments, in this paper we give a self-contained proof that holomorphic algebraic functions verify a uniform Bernstein–Remez inequality. Namely, their growth over a bounded, open, complex set is uniformly controlled by their size on a compact complex subset of sufficiently high cardinality. Up to our knowledge, the first known demonstration on the existence of such an inequality for a specific subset of algebraic functions is contained in Nekhoroshev’s 1973 breakthrough on the genericity of close-to-integrable Hamiltonian systems that are stable over long time. Despite its pivotal rôle, this passage of Nekhoroshev’s proof has remained unnoticed so far. This work aims at extending and generalizing Nekhoroshev’s arguments to a modern framework. We stress the fact that our proof is different from the one contained in Roytwarf and Yomdin’s seminal work (1998), where Bernstein-type inequalities are proved for several classes of functions.