论文标题
具有二次参数的高几何序列的阈值问题
The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters
论文作者
论文摘要
超几何序列是满足一阶线性复发与多项式系数的有理价值序列。也就是说，$ \ langle u_n \ rangle_ {n = 0}^\ infty $如果满足表单$ p（n）u_ {n+1} = q（n）u_ {n）u_ {n）u_ {n} $的一阶线性恢复，则使用polynomial cofficiatient $ u_0 \ in \ mathbb {q} $。 在本文中，我们考虑了超几何序列的阈值问题：给定超测定序列$ \ langle u_n \ rangle_ {n = 0}^\ infty $和一个$ t \ in \ mathbb {q} $的阈值$ t \ in \ mathbb {q} $，确定$ u_n \ ge t $ for $ u_n \ ge t $ n $ n \ n \ n \ n \ n \ n \ in \ in \ in \ in \ in \ n \ n \ in \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n。在假设系数$ p $和$ q $的假设下，我们为阈值问题建立了可决定性。我们还建立了有条理的可决定性结果；例如，在假设系数$ p $和$ q $的假设是一元多项式中，其根源在于$ \ m athbb {q} $的任何数量的二次扩展，阈值问题是Schanuel猜想的真相。最后，我们展示了我们的方法如何恢复并扩展了有关具有二次参数的超几何序列的成员资格问题的最新可决定性结果。
Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, $\langle u_n \rangle_{n=0}^\infty$ is hypergeometric if it satisfies a first-order linear recurrence of the form $p(n)u_{n+1} = q(n)u_{n}$ with polynomial coefficients $p,q\in\mathbb{Z}[x]$ and $u_0\in\mathbb{Q}$. In this paper, we consider the Threshold Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n\rangle_{n=0}^\infty$ and a threshold $t\in\mathbb{Q}$, determine whether $u_n \ge t$ for each $n\in\mathbb{N}_0$. We establish decidability for the Threshold Problem under the assumption that the coefficients $p$ and $q$ are monic polynomials whose roots lie in an imaginary quadratic extension of $\mathbb{Q}$. We also establish conditional decidability results; for example, under the assumption that the coefficients $p$ and $q$ are monic polynomials whose roots lie in any number of quadratic extensions of $\mathbb{Q}$, the Threshold Problem is decidable subject to the truth of Schanuel's conjecture. Finally, we show how our approach both recovers and extends some of the recent decidability results on the Membership Problem for hypergeometric sequences with quadratic parameters.
