Some two variable limit F-signatures
The F-signature is a positive characteristic ring invariant measuring how nice the ring is, on a scale of "not even strongly F-regular" to "regular". One can also consider the F-signature function of a ring element, which carries information about the F-signature, Hilbert-Kunz multiplicity, and F-pure threshold. As a further generalization, given a polynomial in characteristic 0, the limit F-signature function, comes from considering the F-signature mod p, and taking an appropriately scaled limit as the primes p go to infinity. One downside of these invariants is they are quite difficult to compute in practice. In this talk, we will present a formula for the limit F-signature function for a polynomial of the form f=x^ay^b(x+y)^c and we will overview two different ways this can be computed. This is joint work Izzet Coskun, Suchitra Pande, and Kevin Tucker.