Arithmetic in the Boij S\"oderberg Cone

We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-S\"oderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.