D-有限幂级数对角线的单指数界
Single-exponential bounds for diagonals of D-finite power series
AI总结：
本文修正并扩展了Gessel关于D-有限幂级数对角线D-有限性的结果，给出了单指数界的证明。
摘要：
D-有限幂级数在组合数学、数论和数学物理中普遍存在。它们满足一组线性偏微分方程组，其解空间是有限维的，这使得它们享有许多良好的性质。在20世纪80年代他人尝试之后，Lipshitz是第一个证明在多变量情况下，它们所形成的类在对角操作下封闭的。特别是，Gessel之前的工作已经处理了多变量有理幂级数对角线的D-有限性。在本文中，我们给出了Gessel结果的另一种证明，修正了他原始证明中的漏洞，并将其扩展到整个D-有限幂级数的完整类。我们还提供了一个单指数界，该界以输入微分系统的次数和次数来表征D-有限幂级数对角线所满足的定义微分方程的次数和阶数。
英文摘要：
D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.
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