2018/09/04 by Zhaobo, Han, Zhaobo Tom
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1809.01175
openalex publication_date 2018/09/04 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
It was conjectured by Emil Artin in the 1930's that every d-form F(x1, x2, …, xn) over the p-adic field in more than d2 variables has a solution that is not (0, 0, ⋯, 0) (non-trivial solution) over the p-adic field. This is true for d=2 and d=3. However, many counterexamples for d ≥ 4 were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Karačuba Type. The degrees of all known counterexamples, however, are divisible by p-1, which means that they are even for all odd primes. In this article we apply modifications to the known Arkhipov-Karačuba Type counterexamples to construct counterexamples with odd degrees that are divisible by \fracp-12 for all primes greater than 3 and congruent to 3 modulo 4 and then propose some ideas about increasing the number of variables in the counterexamples.