On primary pseudo-polynomials and around Ruzsa's Conjecture

Abstract: Every polynomial $P(X)$ with integer coefficients satisfies the congruences $P(n+m)=P(n) \mod m$ for all integers $n$ and $m$. An integer valued sequence 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 $\limsup |a_n|^{1/n} < e$. A primary pseudo-polynomial is an integer valued sequence $a(n)$ such that $a(n+p)=a(n) \mod p$ for all integers $n ≥ 0$ and all prime numbers $p$. The same conjecture has been formulated for them, which implies Ruzsa’s, and this talk will revolve around this conjecture.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar