论文标题
理性函数的对角线：从差分代数到有效的代数几何（未遗弃版本）
Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)
论文作者
论文摘要
我们表明，我们可以使用创意望远镜的九个和十个参数族的对角线获得的结果，产生的模块化形式可以获得，以折回的$ _2F_1 $超测量功能，可以获得更有效地，从而计算出与eyliptic curve Canonility canonility denominator的$ j $ invariant的计算。如果创意望远镜产生缩减的$ _2F_1 $超几何功能，我们将此结果推广到三个甚至三个以上的变量中的其他有理功能的家庭。当分母与对应于椭圆曲线的产物，椭圆曲线中的叶面的产物相关时，我们还将这一结果推广到三个以上变量中的有理函数。当分母与{\ em属属曲线相关联时，我们还将这些结果扩展到三个变量中的有理函数，使其与两条椭圆形曲线的乘积相对应的jacobian是一个分裂的jacobian}。我们勾勒出理性函数的分母与非一般类型的代数品种相关联，具有无限的生育自动形态。我们最终在三个以上的变量中提供了一些理性功能的示例，其中望远镜已撤回$ _2F_1 $超单几何解决方案，该分母对应于代数品种不仅是椭圆形曲线中的代数曲线，而是在选择的各种椭圆形的曲线中，在各种的曲线中解释了uraldecked $ _2F_2F_1 $ _2f_1 $ _2f_1 $ _2f_1 $ _2fric。
We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained, much more efficiently, calculating the $ j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. In the case where creative telescoping yields pullbacked $ _2F_1$ hypergeometric functions, we generalize this result to other families of rational functions in three, and even more than three, variables. We also generalise this result to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, foliation in elliptic curves. We also extend these results to rational functions in three variables when the denominator is associated with a {\em genus-two curve such that its Jacobian is a split Jacobian} corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked $ _2F_1$ hypergeometric solutions, the denominator corresponding to an algebraic variety being not simply foliated in elliptic curves, but having a selected elliptic curve in the variety explaining the pullbacked $ _2F_1$ hypergeometric solution.
