Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives

Abstract

We first relate an approximate $n^{th}$-order left derivative of $s(R,f^t)$ at the F-pure threshold $c$ to the F-splitting ratio $r_F(R,f^c)$. Next, we apply the methods developed by Monsky and Teixeira in their investigation of syzygy gaps and $p$-fractals to obtain uniform convergence of the F-signature when $f$ is a product of distinct linear polynomials in two variables. Finally, we explicitly compute the F-signature function for several examples using Macaulay2 code outlined in the last section of this paper.

11 pages, 5 figures. Comments and suggestions are welcome. Typos have been fixed and proofs revised

Cited by in corpus (2)

• On the behavior of singularities at the $F$-pure threshold
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