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We initiate the study of the rational SFT capacities of Siegel using tools intoric algebraic geometry. In particular, we derive new (often sharp) bounds forthe RSFT capacities of a strongly convex toric domain in dimension $4$. Thesebounds admit descriptions in terms of both lattice optimization and (toric)algebraic geometry. Applications include (a) an extremely simple latticeformula for for many RSFT capacities of a large class of convex toric domains,(b) new computations of the Gromov width of a class of product symplecticmanifolds and (c) an asymptotics law for the RSFT capacities of all stronglyconvex toric domains.

Lattice Formulas For Rational SFT Capacities