论文标题
在$α\ in \ overline {\ mathbb q}，$和应用程序中评估的二项式系数的总和
Sums of binomial coefficients evaluated at $α\in \overline{\mathbb Q},$ and applications
论文作者
论文摘要
添加剂单体$ r _+（x）$定义为二项式系数的所有非负整数线性组合的集合$ \ binom {x} {n} {n} $ for $ n \ in \ mathbbb z _+$。本文涉及对$ r _+（α）$的构造$α$α的调查。这个问题是由对deligne类别之间的函数的研究$ \ textrm {rep}（s_t）$（以及$ textrm {rep}（\ textrm {rep {pextrm {gl} _t _t）$的$ t \ in \ Mathbb c \ backslash \ backslash \ mathbb z _ _ mathbb z _+$。我们证明，只有$α$是一个不是非负整数的代数数字时，我们才证明该对象是一个环。此外，我们表明，所有代数整数由$α，$，即$ \ MATHCAL O _ {\ MATHBB Q（α）}的所有元素，$也包含在此环中。我们还为代数整数和一般代数数$α提供了两种明确表示的$ r _+（α）$。$一个在某些素数理想方面的不平等方面，而另一种是明确构造的发电机。我们展示了这些结果如何在阳性特征中deligne类别之间的对称单体函数的研究中起作用。此外，这导致对二次代数数和统一根的$ r _+（α）$的特别简单描述。
The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(α)$ for complex numbers $α.$ Particularly interesting is the case of algebraic $α$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbb C\backslash \mathbb Z_+$. We prove that this object is a ring if and only if $α$ is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by $α,$ i.e. all elements of $\mathcal O_{\mathbb Q(α)},$ are also contained in this ring. We also give two explicit representations of $R_+(α)$ for both algebraic integers and general algebraic numbers $α.$ One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of $R_+(α)$ for both quadratic algebraic numbers and roots of unity.
