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F-thresholds are defined by Mustata, Takagi and Watanabe in [F-thresholds andBernstein-Sato polynomials], which are invariants of the pair of ideals onrings of characteristic $p$. In their paper, it is proved F-thresholds equal tojumping numbers for the test ideal on regular local rings. In this note, wegive an formula of F-thresholds on toric rings. This formula is ageneralization of the example in [Huneke, Mustata, Takagi andWatanabe:F-thresholds, tight closure, integral closure, and multiplicitybounds]. We prove that there exists an inequality between F-jumpingcoefficients and F-thresholds. In particular, we observe a comparison betweenF-pure thresholds and F-thresholds. As applications, we prove thecharacterization of regularity for toric rings defined by a simplicial cone,and the rationality of F-thresholds in some cases.

Formulas of F-thresholds and F-jumping coefficients on toric rings