Dual $F$-signatures of Veronese subrings and Segre products of polynomial rings

TL;DR

This paper computes the dual F-signatures of Veronese subrings of polynomial rings using combinatorial methods and provides bounds for Segre products, identifying cases where these bounds are sharp.

Contribution

It introduces explicit calculations of dual F-signatures for Veronese subrings and establishes bounds for Segre products, advancing understanding of their algebraic properties.

Findings

Dual F-signatures of Veronese subrings are explicitly computed.

An upper bound for dual F-signatures of Segre products is established.

The upper bound is shown to be sharp in certain cases.

Abstract

In this paper, we compute the dual $F$-signatures of certain toric rings by using combinatorial techniques. Specifically, we calculate the dual $F$-signatures of Veronese subrings of polynomial rings. Moreover, we give an upper bound for the dual $F$-signatures of Segre products of polynomial rings and show that this upper bound is attained in some cases.