Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11
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arXiv
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| Format: | Preprint |
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2025
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| author | Beneish, Lea Keyes, Christopher |
| author_facet | Beneish, Lea Keyes, Christopher |
| contents | We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a nonreduced $\overline{k}$-point $x$ on $X_f \cap L$, we give an upper bound on the number of planes $P$ containing $L$ for which $X_f \cap P$ contains a line through $x$. Underlying this result is a factorization algorithm for bivariate polynomials originally due to Kaltofen, which we present with slightly relaxed hypotheses. Our primary application is to Artin's conjecture on $p$-adic forms of prime degree $d$: if $K/\mathbb{Q}_p$ is a finite extension with residue field isomorphic to $\mathbb{F}_q$ and $F \in K[x_0, \ldots, x_{d^2}]$ is homogeneous of degree $d$, the conjecture states $F$ has a nontrivial zero in $K$. We show this conjecture holds whenever $q > 679$ for $d=7$ and $q > 7393$ for $d=11$, improving upon a result of Wooley. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11 Beneish, Lea Keyes, Christopher Number Theory 11G20, 11G25, 12E05 We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a nonreduced $\overline{k}$-point $x$ on $X_f \cap L$, we give an upper bound on the number of planes $P$ containing $L$ for which $X_f \cap P$ contains a line through $x$. Underlying this result is a factorization algorithm for bivariate polynomials originally due to Kaltofen, which we present with slightly relaxed hypotheses. Our primary application is to Artin's conjecture on $p$-adic forms of prime degree $d$: if $K/\mathbb{Q}_p$ is a finite extension with residue field isomorphic to $\mathbb{F}_q$ and $F \in K[x_0, \ldots, x_{d^2}]$ is homogeneous of degree $d$, the conjecture states $F$ has a nontrivial zero in $K$. We show this conjecture holds whenever $q > 679$ for $d=7$ and $q > 7393$ for $d=11$, improving upon a result of Wooley. |
| title | Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11 |
| topic | Number Theory 11G20, 11G25, 12E05 |
| url | https://arxiv.org/abs/2508.20192 |