Diagonalization and Rationalization of algebraic Laurent series
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.4090
 Bibcode:
 2012arXiv1205.4090A
 Keywords:

 Mathematics  Number Theory
 EPrint:
 48 pp