Degree bounds have a long history in invariant theory. The Noether bound on the degrees of algebra generators for a ring of invariants is over a century old, and there is a vast literature sharpening and generalizing it. In the last two decades, there has also been an active program on degree bounds for invariants which are able to distinguish orbits as well as algebra generators can (known as separating invariants).

In this talk I make the case that generators for the field of rational invariants represent an exciting avenue for research on degree bounds as well. I present new lower and upper bounds. It will transpire that even the case of G=Z/pZ, uninteresting from the point of view of generating and separating invariants, has a story to tell for rational invariants. The methods involve the classical Minkowski “geometry of numbers”. I argue that this domain of inquiry is both interesting in itself and well-motivated by applications.

This talk is based on joint work with Thays Garcia, Rawin Hidalgo, Consuelo Rodriguez, Alexander Kirillov Jr., Sylvan Crane, Karla Guzman, Alexis Menenses, Maxine Song-Hurewitz, Afonso Bandeira, Joe Kileel, Jonathan Niles-Weed, Amelia Perry, and Alexander Wein.