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This is a Quantitative Study study.

September 24, 2025Open Access

Effective Bertini theorems and zeros of p-adic forms of degree 7

Key Points

An effective Bertini-type theorem establishes bounds on planes intersecting a hypersurface line defined over a finite field.
An upper bound demonstrates that given certain conditions, multiple planes can contain a line through a specified point on the hypersurface.
The results contribute to Artin's conjecture, furthering the understanding of solutions to homogeneous forms of degree 7 in p-adic fields.
The methods involve a factorization algorithm for bivariate polynomials with relaxed hypotheses, offering a more adaptable approach.

Abstract

We establish an effective Bertini-type theorem for hypersurfaces Xf f = 0 defined over a finite field k for which f has no linear factors over the algebraic closure k. Given a line L defined over k and a nonreduced k-point x on Xf L, we give an upper bound on the number of planes P containing L for which Xf 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 degree 7: if K/Qₚ is a finite extension with residue field isomorphic to Fq and F (x₀, , xₙ) Kx₀, , x₄₉ is a homogeneous form of degree 7, then there exists a K-solution to F=0 whenever q>679. This improves on a result of Wooley.

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Cite This Study

Beneish et al. (Wed,) studied this question.

synapsesocial.com/papers/68d6d82e8b2b6861e4c3e327 https://doi.org/https://doi.org/10.48550/arxiv.2508.20192