Symbolic computations in differential geometry

Symbolic integration of hyperexponential 1-forms

Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $ω$ a rational differential $1$-form. Assume that $Hω$ is closed and $H$ transcendental. We prove using Schanuel conjecture that there exist a univariate function $f$ and multivariate rational functions $F,R$ such that $\int Hω= f(F(x))+H(x)R(x)$. We present an algorithm to compute this decomposition. This allows us to present an algorithm to construct a basis of the cohomology of differential $1$-forms with coefficients in $H\mathbb{K}[x…

On Computational Poisson Geometry I: Symbolic Foundations

We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our python module $\textsf{PoissonGeometry}$ implements our algorithms, and accompanies this paper. We include two examples of how our methods can be used, one for gauge transformations of Poisson bivectors in dimension 3, and a second one that determines parametric Poisson bivector fields in dimension 4.

平面微分叶状结构上的符号积分

Symbolic integration on planar differential foliations

本文研究平面微分叶状结构上有理函数的符号积分，提出计算 telescoper 的算法，并证明其存在性或不存在性，应用于寻找平面有理向量场的 Liouvillian 解。