Almost all positive continuous linear functionals can be extended

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.Optimizatio

Divisorial gonality of graphs, the slice rank polynomial method, and tensor products of convex cones

Discrete Mathematics and Optimizatio

On the size of subsets of Fⁿ_q avoiding solutions to linear systems with repeated columns

Consider a system of m balanced linear equations in k variables with coefficients in Fq. 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, …, xk) ∈ Sk with x1, …, xk 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, …, xk are pairwise distinct, or even a solution where x1, …, xk 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.Discrete Mathematics and Optimizatio

Treewidth is a lower bound on graph gonality

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include the grid graphs and the complete multipartite graphs. We prove that the treewidth lower bound also holds for metric graphs (tropical curves) by constructing for any positive rank divisor on a metric graph a positive rank divisor of the same degree on a subdivision of the underlying combinatorial graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al.Optimizatio

Discrete and metric divisorial gonality can be different

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.Optimizatio

Constructing tree decompositions of graphs with bounded gonality

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.Optimizatio