Consider a system of m balanced linear equations in k variables with coefficients in F_q. If k ⩾ 2m + 1, then a routine application of the slice rank method shows that there are constants β, γ ⩾ 1 with γ < q such that, for every subset S ⊆ Fⁿq of size at least β · γⁿ, the system has a solution (x₁, …, x_k) ∈ S^k with x₁, …, x_k not all equal. Building on a series of papers by Mimura and Tokushige and on a paper by Sauermann, this paper investigates the problem of finding a solution of higher non-degeneracy; that is, a solution where x₁, …, x_k are pairwise distinct, or even a solution where x₁, …, x_k do not satisfy any balanced linear equation that is not a linear combination of the equations in the system. In this paper, we focus on linear systems with repeated columns. For a large class of systems of this type, we prove that there are constants β, γ ⩾ 1 with γ < q such that every subset S ⊆ Fⁿq of size at least β · γⁿ contains a solution that is non-degenerate (in one of the two senses described above). This class is disjoint from the class covered by Sauermann’s result, and captures the systems studied by Mimura and Tokushige into a single proof. Moreover, a special case of our results shows that, if S ⊆ Fⁿp is a subset such that S − S does not contain a non-trivial k-term arithmetic progression (with p prime and 3 ⩽ k ⩽ p), then S must have exponentially small density. ... Read more

This dissertation explores three interconnected topics at the interface of combinatorics, algebra, and geometry.

In Part I, we study the divisorial gonality of graphs, a graph parameter inspired by algebraic geometry. Introduced by Baker around 2007, divisorial gonality links chip-firing games on graphs to classical algebraic geometry, including the combinatorial Riemann–Roch theorem and Brill–Noether theory. We contribute in two ways. First, we provide a constructive proof that treewidth is a lower bound for divisorial gonality, presenting a polynomial-time algorithm that transforms a positive rank effective divisor into a tree decomposition of bounded width. This bridges gonality with structural graph theory and enables dynamic programming techniques for graphs of bounded gonality. Second, we examine the Brill–Noether conjecture for graphs and disprove Baker’s subdivision conjecture by constructing graphs whose gonality exceeds that of the associated metric graphs. While the Brill–Noether existence conjecture remains unresolved, our results clarify structural limitations in approaching its proof.

Part II investigates applications of the slice rank polynomial method to affine configurations over finite fields. Extending the method initially developed to solve the cap set problem, we analyze subsets of
Fqn
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q
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avoiding non-trivial solutions to systems of balanced linear equations. For certain classes of systems, we demonstrate the existence of solutions with pairwise distinct or maximally affinely independent elements, generalizing previous results by Mimura and Tokushige. These findings contribute to understanding the limitations and potential of slice rank techniques for broader combinatorial problems.

In Part III, we study tensor products of convex cones, a topic relevant across functional analysis, operator theory, approximation theory, and theoretical physics. Existing literature mainly addresses Archimedean lattice cones or finite-dimensional proper generating cones, leaving many cones unexamined. We extend known results to general cones and present several new contributions: we establish mapping properties for projective/injective cones analogous to norms, characterize their lineality spaces and extremal rays, show containment of projective/injective tensor products in proper cones, and demonstrate that tensoring symmetric convex sets preserves proper faces. For closed cones in finite-dimensional spaces, we show that the projective cone is closed and almost always strictly contained in the injective cone, partially recovering Barker’s conjecture prior to its independent full proof by Aubrun et al. Our work unifies tensor product properties across a wide class of convex cones and provides new tools for both theoretical and applied analysis.

Together, these three parts demonstrate the deep interplay between combinatorics, algebra, and geometry, advancing understanding in graph theory, finite field combinatorics, and convex analysis. The results provide constructive algorithms, generalized theoretical frameworks, and novel counterexamples, opening directions for further research in structural graph theory, algebraic combinatorics, and tensor analysis. ... Read more

Let F be an ordered topological vector space (over R) whose positive cone F₊ is weakly closed, and let E⊆ F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-∗ dense in the topological dual wedge E+′. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces. ... Read more

This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated metric graph Γ(G,1) with unit lengths. We show that dgon(Γ(G,1)) is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker [3, Conjecture 3.14] in the negative. ... Read more

In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality. ... Read more

In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality. ... Read more