Somos 4序列的整性性与Laurent现象
Integrality and the Laurent phenomenon for Somos 4 sequences
AI总结：
本文证明Somos 4序列满足强于Laurent性质的条件，建立整性性的充分条件，并给出非周期序列对应的四元四次丢番图方程的无穷多解，同时推广到Somos 5序列。
摘要：
Somos 4序列是由一个具有常系数的四阶二次递推关系定义的一族序列。对于系数和四个初始数据的特定选择，此类递推可以产生整数序列。Fomin和Zelevinsky利用簇代数理论证明了这些递推也提供了Laurent现象的最简单例子之一：Somos 4序列的所有项都是初始数据的Laurent多项式。某些Somos 4序列的整性性先前已通过Laurent现象得到理解。然而，本文的每位作者都独立建立了Somos 4序列与椭圆曲线上点序列之间的精确对应关系。在此，我们证明这些序列满足比Laurent性质更强的条件，从而建立了整性性的广泛充分条件。作为副产品，非周期序列为相关的四元四次丢番图方程提供了无穷多个解。文中还给出了Somos 5序列的类似结果，以及各种例子，包括Somos 4整数序列的参数族。
英文摘要：
Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.
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