The F-signature of a local ring is a positive characteristic numerical invariant measuring how singular it 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 of the pair. As many of these notions are already analogues of characteristic 0 notions (strong F-regularity connects to klt and F-purity connects to lc), it is reasonable to try a further generalization: given a polynomial g in characteristic 0, the limit F-signature function of g comes from considering the F-signature mod p, and taking the limit as the primes p go to infinity. In this talk, we will present a formula for the limit F-signature function for a binomial of the form g=x^a y^b (x+y)^c and see how this connects with the normalized volume for the divisor div(g^t). Time permitting, we will overview some of the key ideas in computing this formula.

This talk is joint work with Izzet Coskun, Suchitra Pande, and Kevin Tucker.