Diagonalization and Rationalization of algebraic Laurent series

@article{Adamczewski2012DiagonalizationAR,
  title={Diagonalization and Rationalization of algebraic Laurent series},
  author={Boris Adamczewski and Jason P. Bell},
  journal={arXiv: Number Theory},
  year={2012}
}
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… 
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