微分递归、射影判别式与代数生成函数
Differential Recursions, Projective Discriminants, and Algebraic Generating Functions
- Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine（波格柳博夫理论物理研究所，乌克兰国家科学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究由微分递归定义的二元生成函数层级，揭示其代数性质与Galois群，提出代数性猜想并识别超越性机制。
AI中文摘要：
我们研究由微分递归定义的两类二元生成函数层级：自对偶层级和约化对称层级。它们的有理系数函数产生回文和反回文三角形，包括OEIS A336110和A140136。对于一般微分阶数$p$，约化层级分解为$p-1$个Horn型伴随函数。一个显式的$p-3$阶微分交织子将最后一个伴随函数与归一化自对偶生成函数联系起来。这些伴随函数共享一个主符号和一条具有射影$S_3$作用的有理参数化特征曲线。当$p=2$时，该构造产生Narayana二次式。当$p=3$时，对称性提升为$S_4$：两个代数扇区是一个四次根配置的边预解式，非对称Gross--Witten--Wadia生成函数的次数为四。对于$p=4$和$p=5$，我们分别推导出次数为六和八的代数方程，并在一般特殊化下找到Galois群$S_6$和$S_8$。当$p=6$时，我们验证了一个次数为十的特殊化，其Galois群为$S_{10}$。它们的判别式表现出一个普遍的三次特征因子和一个平方的低阶因子。边界分解解释了观察到的代数次数和判别式单项式。我们对约化层级中的代数性和Galois群提出猜想，并识别出更高自对偶层级中超越性的一种可能机制。
英文摘要：
We study two hierarchies of bivariate generating functions defined by differential recursions: a self-dual hierarchy and a reduced-symmetry hierarchy. Their rational coefficient functions produce palindromic and anti-palindromic triangles, including OEIS A336110 and A140136. For general differential order $p$, the reduced hierarchy decomposes into $p-1$ Horn-type companions. An explicit differential intertwiner of order $p-3$ relates the last companion to the normalized self-dual generating function. The companions share a principal symbol and a rationally parametrized characteristic curve with a projective $S_3$ action. At $p=2$ this construction yields the Narayana quadratic. At $p=3$ the symmetry lifts to $S_4$: two algebraic sectors are edge resolvents of one quartic root configuration, and the nonsymmetric Gross--Witten--Wadia generating function has degree four. For $p=4$ and $p=5$ we derive algebraic equations of degrees six and eight, respectively, and find Galois groups $S_6$ and $S_8$ at generic specializations. At $p=6$ we verify a degree-ten specialization with Galois group $S_{10}$. Their discriminants exhibit a universal cubic characteristic factor and a squared lower-order factor. A boundary factorization explains the observed algebraic degrees and discriminant monomials. We formulate conjectures on algebraicity and Galois groups in the reduced hierarchy and identify a possible mechanism for transcendence in the higher self-dual hierarchy.
