On primary pseudo-polynomials (around Ruzsa’s conjecture)

Abstract

Every polynomial [Formula: see text] satisfies the congruences [Formula: see text] for all integers [Formula: see text]. An integer valued sequence [Formula: see text] is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long-standing conjecture of Ruzsa says that a pseudo-polynomial [Formula: see text] is a polynomial as soon as [Formula: see text]. Under this growth assumption, Perelli and Zannier proved that the generating series [Formula: see text] is a [Formula: see text]-function. A primary pseudo-polynomial is an integer valued sequence [Formula: see text] such that [Formula: see text] for all integers [Formula: see text] and all prime numbers [Formula: see text]. The same conjecture has been formulated for them, which implies Ruzsa’s, and this paper revolves around this conjecture. We obtain a Hall type characterization of primary pseudo-polynomials. We give a new proof and generalize a result due to Zannier that any primary pseudo-polynomial with an algebraic generating series is a polynomial. We make the Perelli–Zannier Theorem effective and we prove a Pólya type result.