A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations
We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.
Keywordsalgebraic equations Mellin–Barnes integral hypergeometric functions convergence domain
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