Diagonals and algebraicity modulo $p$: a sharper degree bound
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917215297601536 |
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| author | Adamczewski, Boris Bostan, Alin Caruso, Xavier |
| author_facet | Adamczewski, Boris Bostan, Alin Caruso, Xavier |
| contents | In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in $\mathbb F_p$. Moreover, he conjectured that the algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on $d_p$ with an explicit and reasonable degree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14920 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Diagonals and algebraicity modulo $p$: a sharper degree bound Adamczewski, Boris Bostan, Alin Caruso, Xavier Symbolic Computation Number Theory In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in $\mathbb F_p$. Moreover, he conjectured that the algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on $d_p$ with an explicit and reasonable degree. |
| title | Diagonals and algebraicity modulo $p$: a sharper degree bound |
| topic | Symbolic Computation Number Theory |
| url | https://arxiv.org/abs/2601.14920 |