# Symbolic integration of hyperexponential 1-forms

## TL;DR

This paper develops an algorithmic approach to integrate hyperexponential 1-forms symbolically, leveraging the Schanuel conjecture, and applies it to construct cohomology bases and linearize certain differential systems.

## Contribution

It introduces a novel algorithm for decomposing integrals of hyperexponential forms and constructing cohomology bases, extending the understanding of differential equations with Liouvillian first integrals.

## Key findings

- Algorithm computes integral decompositions involving univariate functions.
- Constructs bases for cohomology of differential 1-forms with hyperexponential coefficients.
- Generalizes Singer's result by providing rational transformations to linearize certain systems.

## Abstract

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 $\omega$ a rational differential $1$-form. Assume that $H\omega$ 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 $\int H\omega= f(F(x))+H(x)R(x)$. 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 $H\mathbb{K}[x,1/(SD)]$ for a given $H$, $D$ being the denominator of $dH/H$ and $S\in\mathbb{K}[x]$ a square free polynomial. As an application, we generalize a result of Singer on differential equations on the plane: whenever it admits a Liouvillian first integral $I$ but no Darbouxian first integral, our algorithm gives a rational variable change linearising the system.