STNB2025 (38th edition)
Images of certain p-adic polynomials, their ratio sets, and a conjecture in additive combinatorics
Ponentes
Stevan Gajovic
Resumen
For a given polynomial $f$ in $\mathbb{Z}_p[x]$, we consider the question if the ratio set $f(x)/f(y)$, where $x$ and $y$ are in $\mathbb{Z}_p$, and $f(y)$ is not zero, is equal to $\mathbb{Q}_p$. Miska, Murru, and Sanna proved that the answer is yes if f has a simple root or, more generally, if $f$ has two roots with coprime multiplicities. Let $q>1$ be an integer. We restrict our attention to polynomials that are a product of a $q$th power of a polynomial and a product of irreducible polynomials whose degrees are divisible by $q$. We give a criterion for when the answer to the starting question is no, and we give examples when the ratio sets are equal to $\mathbb{Q}_p$ and discuss the question of the minimal number of such factors; this is related to a conjecture in additive combinatorics. We apply our statements to give a criterion for polynomials of small degree. This is joint work with Deepa Antony and Rupam Barman.
Ficheros
No hay ficheros disponibles para descargar
