Tight universal quadratic forms

arXiv2104.02440MaRDI QIDQ6364639

Min-Gyu Kim, Byeong-Kweon Oh

Publication date: 6 April 2021

Abstract: For a positive integer $n$, let $mathcalT(n)$ be the set of all integers greater than or equal to $n$. An integral quadratic form $f$ is called tight $mathcalT(n)$-universal if the set of nonzero integers that are represented by $f$ is exactly $mathcalT(n)$. The smallest possible rank over all tight $mathcalT(n)$-universal quadratic forms is defined by $t(n)$. In this article, we find all tight $mathcalT(n)$-universal diagonal quadratic forms. We also prove that $t(n)inOmega(lo{g}_2(n))capO(sqrtn)$. Explicit lower and upper bounds for $t(n)$ will be provided for some small integer $n$.

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