# Ax-Schanuel for $GL_n$

## TL;DR

This paper establishes an Ax-Schanuel type theorem for exponential functions on general linear groups over complex numbers, extending known results and deriving corollaries related to bi-algebraic subsets.

## Contribution

It proves the Ax-Schanuel theorem for $GL_n(C)$ and its subgroup of upper triangular matrices, providing new insights into their algebraic and transcendental properties.

## Key findings

- Proved Ax-Schanuel for upper triangular matrices
- Extended Ax-Schanuel to all $GL_n(C)$
- Derived Ax-Lindemann type results and characterized bi-algebraic subsets

## Abstract

In this paper we prove an Ax-Schanuel type result for the exponential functions for general linear groups over $\mathbb{C}$. We prove the result first for the group of upper triangular matrices and then for the group $GL_n$ of all $n\times n$ invertible matrices over $\mathbb{C}$. We also obtain Ax-Lindemann type results for these maps as a corollary, characterizing the bi-algebraic subsets of these maps.