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Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11

Lea Beneish, Christopher Keyes

Abstract

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.

Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11

Abstract

We establish an effective Bertini-type theorem for hypersurfaces defined over a finite field for which has no linear factors over the algebraic closure . Given a line defined over and a nonreduced -point on , we give an upper bound on the number of planes containing for which contains a line through . 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 -adic forms of prime degree : if is a finite extension with residue field isomorphic to and is homogeneous of degree , the conjecture states has a nontrivial zero in . We show this conjecture holds whenever for and for , improving upon a result of Wooley.

Paper Structure

This paper contains 15 sections, 22 theorems, 45 equations, 1 table.

Table of Contents

  1. Introduction
  2. On Bertini theorems
  3. On Artin's conjecture
  4. Notation and organization
  5. Effective Bertini theorems
  6. A factorization algorithm
  7. Genericity conditions
  8. Proofs of Theorem \ref{['thm:effBertini']} and \ref{['thm:effBertini_existence']}
  9. Degree d forms over Fq
  10. d=7
  11. d=11
  12. Reduced forms and consequences
  13. Reduced forms
  14. Consequences
  15. Proof of Theorem \ref{['thm:deg711']}

Key Result

Theorem A

Let $d \geq 2$, $n \geq 3$, and $q > {2d^2}$. Suppose $f \in \mathbb{F}_q[x_0,\ldots,x_n]$ is a degree $d$ form, irreducible over $\overline{\mathbb{F}_q}$, and $X_f \colon f = 0$ is the associated geometrically integral hypersurface in $\mathbb{P}^n$ defined over $\mathbb{F}_q$.

Theorems & Definitions (48)

  • Theorem A: effective Bertini irreducibility for planes, existence form (see Theorem \ref{['thm:effBertini_existence']})
  • Conjecture : Artin's conjecture for prime degree $p$-adic forms
  • Theorem B
  • Theorem 2.1: effective Bertini irreducibility for planes
  • Theorem 2.2: effective Bertini irreducibility for planes, existence form
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6: $D=1$
  • Proposition 2.7
  • ...and 38 more