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

Technical Summary: The Hasse Principle for Diagonal Forms Restricted to a Hypersurface of Adjacent Degree

Problem Statement
The paper investigates the Hasse principle for Diophantine systems consisting of two homogeneous forms, $F$ and $G$, in $n$ variables over the integers. Specifically, the study focuses on the case where $F$ is a diagonal form of degree $k \ge 3$ and $G$ is a general (non-diagonal) form of degree $d = k-1$. The primary objective is to establish an asymptotic formula for the number of integer solutions, denoted $N_{F,G}(X)$, within the box $[-X, X]^n$ satisfying $F(\mathbf{x}) = G(\mathbf{x}) = 0$.

The expected asymptotic behavior is of the form:
$$N_{F,G}(X) = X^{n-k-d} \left( C_{F,G} + O(X^{-\nu}) \right)$$
where $C_{F,G} \ge 0$ is a constant representing the product of local densities (singular series and singular integral), and $\nu > 0$. The central challenge addressed is determining the minimal number of variables $n_0(k, d)$ required to guarantee this asymptotic formula for non-singular forms.

Methodology
The proof utilizes the Hardy-Littlewood circle method, refining the approach established by Brandes and Parsell [BP21] for systems where one form is diagonal. The argument proceeds through the following technical steps:

1. Circle Method Setup: The counting function is expressed as an integral over the unit torus $\mathbb{T}^2$ of the exponential sum $S(\alpha) = \sum_{\mathbf{x}} e(\alpha_k F(\mathbf{x}) + \alpha_d G(\mathbf{x}))$. The domain is partitioned into major arcs ($\mathcal{N}$) and minor arcs ($\mathfrak{n}$).
2. Minor Arc Analysis (One-Dimensional): The authors first address the case where $\alpha_k$ (the frequency associated with the diagonal form) is not well-approximated by rationals. By employing $d$ successive Weyl differencing steps, the degree $d$ term is eliminated, reducing the problem to estimating exponential sums involving linear polynomials. Standard Weyl estimates and divisor bounds are applied to show that the contribution from these minor arcs is negligible provided $n$ is sufficiently large.
3. Minor Arc Analysis (Two-Dimensional): When $\alpha_k$ is well-approximated by a rational $a/q$, the analysis shifts to $\alpha_d$. Using a two-dimensional splitting of the major arcs, the authors demonstrate that unless $\alpha_d$ is also well-approximated by a rational with the same denominator (or a related one), the exponential sum remains small. This step relies on a refined application of Weyl differencing and specific bounds on the size of the exponential sum derived from the diagonal structure of $F$.
4. Major Arc Analysis: On the major arcs, where both $\alpha_k$ and $\alpha_d$ are close to rationals, the integral is approximated by a product of a singular series (arising from the discrete sum over residues) and a singular integral (arising from the continuous integral over the real domain). The convergence of these series and integrals is established under the derived bounds on $n$.

Key Contributions and Results
The paper's main contribution is the improvement of the upper bound for the number of variables $n$ required to satisfy the Hasse principle in the specific case of adjacent degrees ($d = k-1$).

• Previous Bounds:

  • For general forms (no diagonality), Browning and Heath-Brown [BH17] provided a bound growing exponentially in $k$ and $d$.
  • For one diagonal form $F$ (degree $k$) and one general form $G$ (degree $d < k$), Brandes and Parsell [BP21] improved this to a quadratic bound in $k$. Specifically, for $d = k-1$, they established $n_0(k, k-1) \le 2^k k$.
  • For the specific case of a cubic and a quadratic ($k=3, d=2$), the bound was $n_0(3, 2) \le 24$.

• New Result (Theorem 1.1):
  The author refines the use of the Weyl inequality in the specific setting of adjacent degrees to prove that the asymptotic formula holds if:
  $$n > 2^{k-1}(2k - 1)$$
  This represents a strict improvement over the $2^k k$ bound from [BP21].

• Corollary 1.2 (Cubic and Quadratic):
  Applying the main theorem to the case $k=3$ and $d=2$, the requirement is relaxed from $n > 24$ to:
  $$n > 20$$
  This reduces the number of variables needed by four compared to the previous best result for this specific system.

Significance and Scope
The paper claims significance primarily in its technical refinement of existing circle method arguments. By streamlining the application of Weyl inequalities for the specific case of $d = k-1$, the author achieves a tighter bound on the number of variables without introducing new, complex machinery. The argument is presented as largely self-contained, relying on standard results from the circle method and Weyl differencing, making it an accessible entry point for understanding the Hasse principle in this context.

The author explicitly notes that while the methods could theoretically be extended to $d < k-1$, the resulting bound $n > 2^{k-1}(k+d)$ would be weaker than the existing general bound from [BP21] for $d \le k-3$. Therefore, the improvement is specific to the "adjacent degree" case. The paper does not propose new applications or future experimental directions but focuses on sharpening the theoretical threshold for the validity of the asymptotic formula.