A $p$-adic variant of Kontsevich-Zagier integral operation rules and of Hrushovski-Kazhdan style motivic integration

TL;DR

This paper demonstrates that $p$-adic measures of semi-algebraic sets can be derived solely from basic integral rules, linking $p$-adic and motivic integration and addressing a question posed by Kontsevich-Zagier.

Contribution

It establishes that $p$-adic measure equality can be deduced from fundamental rules, showing that universal motivic integration over $Q_p$ aligns with $p$-adic integration.

Findings

$p$-adic measure equality follows from basic integral rules

Universal motivic integration over $Q_p$ is equivalent to $p$-adic integration

Addresses a $p$-adic analogue of a question by Kontsevich-Zagier

Abstract

We prove that if two semi-algebraic subsets of $\mathbb{Q}_p^n$ have the same $p$-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a $p$-adic analogue of a question asked by Kontsevich-Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over $\mathbb{Q}_p$, universal motivic integration (in the sense of Hrushovski-Kazhdan) is just $p$-adic integration.