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arXiv · 2208.03304

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

Christian Porter·Andrew Mendelsohn
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Abstract

Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $\Delta_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by $O( \Delta_K \exp(2n \log(n)+f(n,R_K)))$, where $f(n,R_K)$ is a finite value, satisfying $f(n,R_K)=\frac{\sqrt{n-1}}{2}R_K^{\frac{1}{n-1}}+\frac{4}{n-1}\log(\sqrt{|\Delta_K|})^2$ if $n \leq 11$. Moreover, if $K$ is a unit reducible field, the number of classes of perfect unary forms is bound above by $O( \Delta_K \exp(2n \log(n)))$.

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Christian Porter, Andrew Mendelsohn. 2022-08-05. An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields. https://arxiv.org/abs/2208.03304

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