A Quantitative Hasse Principle for Weighted Quartic Forms

2023/10/06 by Daniel Flores, Flores, Daniel
Mathematics · #11D45 #11D72 #11E76 #11L07 #11P55 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.06868

openalex publication_date 2023/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials F(\mathbf x,\mathbf y) ∈ ℤ[x₁,…,x_s1,y₁,…,y_s2] satisfy the condition that F(λ² \mathbf x, λ\mathbf y) = λ⁴ F(\mathbf x,\mathbf y). Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when s₁ ≥ 2 and 2s₁ + s₂ > 8.

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