Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypersurfaces passing through the Galois orbit of a point

Shamil Asgarli, Jonathan Love, Chi Hoi Yip

TL;DR

The paper resolves the open case $|K|=2$ of a problem on hypersurfaces by proving that for any separable extension $L/K$ of degree $r$ there exists $P\uparrow ext{$P^n(L)$}$ with the vanishing-dimension of degree $d$ forms exactly $ obreak obreak obreak max(m-r,0)$, where $m=inom{n+d}{n}$. It develops a unified framework combining infinite-field general-position arguments and three complementary finite-field methods—incidence counting, irreducible-component counting, and inclusion–exclusion—to obtain positive probability of such a point and to bound the exceptional cases, with computational verification for the remaining small parameter sets. The work also provides natural generalizations to linear systems of hypersurfaces with prescribed factor- or reduced-ness properties, giving sharp dimension bounds and exact results in several cases (including a broader finite-field proof of AGR24). These results illuminate the structure of hypersurfaces through Galois orbits and yield practical tools for constructing linear systems with targeted geometric properties, across both finite and infinite base fields. The techniques combine Lang–Weil type point counts, minor-determinant encodings, component-wise counting, and refined singularity analysis, highlighting deep connections between arithmetic geometry and the geometry of linear systems.

Abstract

Asgarli, Ghioca, and Reichstein recently proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.

Hypersurfaces passing through the Galois orbit of a point

TL;DR

The paper resolves the open case of a problem on hypersurfaces by proving that for any separable extension of degree there exists P^n(L) with the vanishing-dimension of degree forms exactly , where . It develops a unified framework combining infinite-field general-position arguments and three complementary finite-field methods—incidence counting, irreducible-component counting, and inclusion–exclusion—to obtain positive probability of such a point and to bound the exceptional cases, with computational verification for the remaining small parameter sets. The work also provides natural generalizations to linear systems of hypersurfaces with prescribed factor- or reduced-ness properties, giving sharp dimension bounds and exact results in several cases (including a broader finite-field proof of AGR24). These results illuminate the structure of hypersurfaces through Galois orbits and yield practical tools for constructing linear systems with targeted geometric properties, across both finite and infinite base fields. The techniques combine Lang–Weil type point counts, minor-determinant encodings, component-wise counting, and refined singularity analysis, highlighting deep connections between arithmetic geometry and the geometry of linear systems.

Abstract

Asgarli, Ghioca, and Reichstein recently proved that if is a field with , then for any positive integers and , and separable field extension with degree , there exists a point which does not lie on any degree hypersurface defined over . They asked whether the result holds when . We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer and separable field extension with degree , there exists a point such that the vector space of degree forms over that vanish at has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.
Paper Structure (17 sections, 27 theorems, 105 equations, 1 figure)

This paper contains 17 sections, 27 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Applications to linear systems
  3. Proof outline of Theorem \ref{['thm:main']}
  4. Proof for infinite fields
  5. Preliminaries and setup for finite fields
  6. Special cases and reductions
  7. A useful lemma from calculus
  8. Bounds on the number of rational points on a hypersurface
  9. Bounds on the number of reducible hypersurfaces
  10. First method: incidence correspondence
  11. Checking the inequality
  12. Second method: counting by irreducible component
  13. Third method: inclusion-exclusion
  14. Bounds on reducible intersections of hypersurfaces
  15. Inclusion-exclusion
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K$ be a field with $|K| > 2$, and let $d, n$ be positive integers. For any separable field extension $L/K$ with degree $m = \binom{n + d}{d}$, there exists a point $P \in \mathbb{P}^n(L)$ that does not lie on any degree $d$ hypersurface defined over $K$.

Figures (1)

  • Figure 1: How to prove Theorem \ref{['thm:main']} for $K=\mathbb{F}_2$, each given $(n,d)$, and $r=m=\binom{n+d}{n}$.

Theorems & Definitions (57)

  • Theorem 1.1: Asgarli-Ghioca-Reichstein
  • Theorem 1.2
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']} when $K$ is infinite
  • Lemma 3.1
  • proof
  • ...and 47 more