Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in\n at least five Variables

2018/10/28 by Paul Buterus, Friedrich Götze, Buterus, Paul +3
Mathematics · #11D75 #11J25 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1810.11898

openalex publication_date 2018/10/28 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

For any \ε > 0 we derive effective estimates for the size of a\nnon-zero integral point m \∈ \ℤ^d \∖ 0 solving the\nDiophantine inequality \| Q[m] \| < \ε, where Q[m] = q₁\nm₁² + \… + q_d m_d² denotes a non-singular indefinite diagonal\nquadratic form in d \≥ 5 variables. In order to prove our quantitative\nvariant of the Oppenheim conjecture, we extend an approach developed by Birch\nand Davenport [BD58b] to higher dimensions combined with a theorem of\nSchlickewei [Sch85]. The result obtained is an optimal extension of\nSchlickewei's result, giving bounds on small zeros of integral quadratic forms\ndepending on the signature (r,s), to diagonal forms up to a negligible growth\nfactor.\n

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