On Christol's conjecture

Abstract

We show that the unresolved examples of Christol's conjecture ${}_3{F}_2\left([2/9,5/9,8/9],[2/3,1],x\right)$ and ${}_3{F}_2\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other ${}_3{F}_2$ and ${}_4{F}_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the ${}_3{F}_2([1/9,4/9,5/9],[1/3,1],27\cdot x)$ function is a diagonal of a rational function.

13 pages