Diagonalization and Rationalization of algebraic Laurent series

TL;DR

This paper provides a quantitative proof that the diagonal of a multivariate algebraic power series over a field of positive characteristic remains algebraic after reduction modulo a prime, with explicit bounds on degree and height.

Contribution

It offers a quantitative version of a classical result, giving explicit bounds on algebraic properties of diagonals after reduction modulo primes.

Findings

Diagonal of algebraic power series remains algebraic modulo p

Explicit bounds on degree and height after reduction

Answers a question posed by Deligne

Abstract

We prove a quantitative version of a result of Furstenberg and Deligne stating that the the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^{A}$ and height at most $A^2p^{A+1}$, where $A$ is an effective constant that only depends on the number of variables, the degree of $f$ and the height of $f$. This answers a question raised by Deligne.