Solubility of Additive Forms of Twice Odd Degree over ℚ₂(√(5))

2022/07/19 by Drew Duncan, Duncan, Drew, David B. Leep +1
Mathematics · #11D72 #11D88 #11E76 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Computer science #Degree (music) #Discrete mathematics #Extension (predicate logic) #FOS: Mathematics #Geometry #Limits and Structures in Graph Theory #Mathematics #Number Theory (math.NT) #Order (exchange) #Physics #Quadratic equation #Zero (linguistics)

paper · pdf · doi:10.48550/arxiv.2207.09556

openalex publication_date 2022/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that an additive form of degree d=2m, m odd, m≥3, over the unramified quadratic extension ℚ₂(√(5)) has a nontrivial zero if the number of variables s satisifies s ≥ 4d+1. If 3 \nmid d, then there exists a nontrivial zero if s ≥ (3)/(2)d + 1, this bound being optimal. We give examples of forms in 3d variables without a nontrivial zero in case that 3 | d.

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