No 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.Theorem A (see Theorem 3.4). Let K be a perfect field of prime characteristic p. Let R 1 = K[x 1 , . . . , x n ] and R 2 = K[y 1 , . . . , y m ] with maximal homogeneous ideals m 1 = (x 1 , . . . , x n ) and

We show some examples of topological zeta functions associated to an isolated plane curve singular point and an allowed, in the sense due to Némethi and Veys in 2012, differential form that have several poles of order two. This is in contrast to the case of the standard differential form where, as showed by Veys for plane curves in 1995, and by Nicaise and Xu in general twenty one years later, there is always at most one pole of maximal order.