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We give an axiomatic characterization of multiple Dirichlet series over thefunction field $\mathbb F_q(T)$, generalizing a set of axioms given by Diaconuand Pasol. The key axiom, relating the coefficients at prime powers to sums ofthe coefficients, formalizes an observation of Chinta. The existence ofmultiple Dirichlet series satisfying these axioms is proved by exhibiting thecoefficients as trace functions of explicit perverse sheaves, and usingproperties of perverse sheaves. The multiple Dirichlet series defined this wayinclude, as a special case, many that have appeared previously in theliterature.

General multiple Dirichlet series from perverse sheaves