On primary pseudo-polynomials (Around Ruzsa's Conjecture)

Every polynomial $P(X)\in \mathbb Z[X]$ satisfies the congruences $P(n+m)\equiv P(n) \mod m$ for all integers $n, m\ge 0$. An integer valued sequence $(a_n)_{n\ge 0}$ 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 $a_n$ is a polynomial as soon as $\limsup_n \vert a_n\vert^{1/n}<e$. Under this growth assumption, Perelli and Zannier proved that the generating series $\sum_{n=0}^\infty a_n z^n$ is a $G$-function. A primary pseudo-polynomial is an integer valued sequence $(a_n)_{n\ge 0}$ such that $a_{n+p}\equiv a_n \mod p$ for all integers $n\ge 0$ and all prime numbers $p$. 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 and draw various consequences from it. 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. This leads us to formulate a conjecture on diagonals of rational fractions and primary pseudo-polynomials, which is related to classic conjectures of Christol and van der Poorten. We make the Perelli-Zannier Theorem effective. We prove a P\'olya type result: if there exists a function $F$ analytic in a right-half plane with not too large exponential growth (in a precise sense) and such that for all large $n$ the primary pseudo-polynomial $a_n=F(n)$, then $a_n$ is a polynomial. Finally, we show how to construct a non-polynomial primary pseudo-polynomial starting from any primary pseudo-polynomial generated by a $G$-function different of $1/(1-x)$.