论文标题
部分可观测时空混沌系统的无模型预测
On Abel's problem and Gauss congruences
论文作者
论文摘要
亚伯引起的经典问题是确定微分方程$ y'=ηy$是否在$ \ mathbb c（x）$上允许$ y $ y $代数$ y $ y $ y $ y $ h $ h $ h $ hum $η$是$ \ mathbb c（x）$的给定代数函数时。 Risch设计了一种算法，鉴于$η$，它决定是否存在代数解决方案。在本文中，当$η$在$ \ mathbb c \ c \ cup \ {\ infty \} $的某个时刻接受PUISEUX扩展时，我们会采用不同的观点，可以假定为0而不会损失一般性。我们证明了以下算术表征：当时存在$ y'=ηy$的非平凡代数解决方案，并且仅当Puiseux扩展的系数$xη（x）$ at $ 0 $ $ 0 $满足Gauss的一致性几乎所有质量数字。然后，我们将标准应用于高几何体系列：当$xη（x）$是具有有理参数的代数超几何序列时，我们完全确定了带有代数解决方案的方程式$ y'=ηy$，这使我们能够证明使用动机理论进行预测Golyshev。我们还提出了其他三个应用，特别是针对理性分数的对角线和指示二维步行。
A classical problem due to Abel is to determine if a differential equation $y'=ηy$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $η$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $η$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $η$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=ηy$ if and only if the coefficients of the Puiseux expansion of $xη(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=ηy$ with an algebraic solution when $xη(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks.
