$F$-thresholds and test ideals of Thom-Sebastiani type polynomials

TL;DR

This paper derives a formula for F-thresholds of Thom-Sebastiani polynomials over perfect fields of prime characteristic, extending known results for diagonal hypersurfaces, and computes their first test ideals, with applications to log canonical thresholds.

Contribution

It introduces a new formula for F-thresholds of Thom-Sebastiani polynomials and computes their initial test ideals, expanding understanding of singularity invariants in prime characteristic.

Findings

Derived a formula for F-thresholds of Thom-Sebastiani polynomials

Computed the first test ideal for these polynomials

Identified hypersurfaces where log canonical thresholds equal F-pure thresholds for infinitely many primes

Abstract

We provide a formula for $F$-thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the $F$-pure threshold of a diagonal hypersurface. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our result to find hypersurfaces where the log canonical thresholds equals the $F$-pure thresholds for infinitely many prime numbers.