We don't have a nice theory of fields with two (or more) commuting automorphisms, but there are interesting commuting automorphisms out there. For example, fields of rational functions in one variable $t$ have a whole commmuting family of endomorphisms $\sigma_n$ given by $t\mapsto t^n$ for integers $n$; and so do fields of Laurent series in $t$. On algebraic closures of these fields, these endomorphisms become automorphisms. Mahler thought a lot about polynomial difference equations in this setting a long time ago. Several recent results show that certain difference equations for $\sigma_n$ for different $n$ are incompatible in this context. Some of these results rely on some ideas from the model theory of fields with one automorphism, and that's what this talk is about.

This is joint work with Khoa Nguyen and Thomas Scanlon.