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The Hasse principle for diagonal forms restricted to a hypersurface of adjacent degree

Anna Theorin Johansson

TL;DR

Using the Hardy-Littlewood circle method, the paper analyzes the Hasse principle for a Diophantine system with a diagonal form $F$ of degree $k$ and a form $G$ of degree $k-1$. The authors refine Weyl-type estimates and implement a two-stage major/minor arc analysis to obtain an asymptotic count for $N_{F,G}(X)$ with a main term given by a product of a real singular integral and $p$-adic densities, under the condition $n>2^{k-1}(2k-1)$. This yields a concrete improvement over prior bounds, notably giving $n>20$ for the cubic-plus-quadratic case, by reducing the required dimensional threshold from Brandes-Parsell. The result provides a self-contained, accessible circle-method framework for this restricted mixed-degree setting, delivering both convergence of the local densities and a positive singular product under non-singular local solubility.

Abstract

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $k-1$. By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we improve the bound $n > 2^k k$ to $n > 2^{k-1}(2k-1)$; in particular, the requirement $n > 24$ in the case of degrees three and two is relaxed to $n > 20$.

The Hasse principle for diagonal forms restricted to a hypersurface of adjacent degree

TL;DR

Using the Hardy-Littlewood circle method, the paper analyzes the Hasse principle for a Diophantine system with a diagonal form of degree and a form of degree . The authors refine Weyl-type estimates and implement a two-stage major/minor arc analysis to obtain an asymptotic count for with a main term given by a product of a real singular integral and -adic densities, under the condition . This yields a concrete improvement over prior bounds, notably giving for the cubic-plus-quadratic case, by reducing the required dimensional threshold from Brandes-Parsell. The result provides a self-contained, accessible circle-method framework for this restricted mixed-degree setting, delivering both convergence of the local densities and a positive singular product under non-singular local solubility.

Abstract

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree and one general form of degree . By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we improve the bound to ; in particular, the requirement in the case of degrees three and two is relaxed to .
Paper Structure (11 sections, 11 theorems, 88 equations)

This paper contains 11 sections, 11 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. Notation
  4. Overview of paper
  5. Minor arc contribution
  6. Major arc contribution
  7. Weyl estimates
  8. Minor arc contribution
  9. Two-dimensional split
  10. Two-dimensional minor arcs
  11. Major arc contribution

Key Result

Theorem 1.1

Let $F, G \in \mathop{\mathrm{\mathbb{Z}}}\nolimits[x_1, \dots, x_n]$ be a pair of non-singular forms, where $F$ is diagonal of degree $k \geq 3$ of the shape and $G$ has degree Suppose that Then for some $\nu > 0$, we have where $\mathop{\mathrm{\mathcal{C}}}\nolimits_{F, G} \geq 0$ is a product of local solution densities associated with the system $F(\mathop{\mathrm{\mathbf{x}}}\nolimits) =

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 18 more