Lovász-Saks-Schrijver ideals and the dual F-signature of Veronese subrings.

This dissertation presents two independent projects that demonstrate the interplay between commutative algebra and combinatorics. The first project delves into the study of certain algebraic properties of Lovász-Saks-Schrijver ideals and rings. Every simple finite graph G has an associated Lovász-Saks-Schrijver ring R_G(d) that is related to the d-dimensional orthogonal representations of G. We present a link between algebraic properties such as normality, factoriality and strong F-regularity of R_G(d) and combinatorial invariants of the graph G. In particular we prove that if d ≥ pmd(G)+k(G) then R_G(d) is F-regular in positive characteristic and rational singularity in characteristic 0 and furthermore if d ≥ pmd(G)+k(G)+1 then R_G(d) is a UFD. Here pmd(G) is the positive matching decomposition number of G and k(G) is its degeneracy. In the second project, we explore the dual F-signature of rings, a numerical invariant used to study singularities in positive characteristic. This invariant detects F-regularity and F-rationality, two important algebraic properties in singularity theory over prime characteristic fields. While few examples of dual F-signature computations exist, we calculate the dual F-signature of Veronese subrings of polynomial rings in d variables. Although our approach is primarily algebraic, the same result can also be derived using combinatorial tools, showcasing the deep connection between algebra and combinatorics.