A system of n algebraic equations for n unknowns is considered, in which the collection of exponents is fixed, and the coefficients are variable. Since the solutions of such systems are 2n-homogeneous, two coefficients in each equation can be fixed, which makes it possible to pass to the corresponding reduced systems. For the reduced systems, a formula for the solution (and also for any monomial of the solution) is obtained in the form of a hypergeometric type series in the coefficients. Such series are represented as a finite sum of Horn's hypergeometric series: the ratios of the neighboring coefficients of the latter series are rational functions of summation variables. The study is based on the linearization procedure and on the theory of multidimensional residues. As an application of the main formula, a multidimensional analog is presented of the Waring formula for powers of the roots of the system.

In the paper we obtain the criterion for the convergence of a Mellin-Barnes integral representing a solution to a general system of algebraic equations. This result yields also a criterion to principal minors of a matrix with nonnegative elements to be positive. The proof is based on the theorem on domains of convergence of Mellin-Barnes integrals by Nilsson-Passare-Tsikh and the wellknown theorem of linear algebra on a decomposition of real space into polyhedral cones.

In the paper, we deal with the problem of getting analytic continuations for the monomial function associated with a solution to the reduced trinomial algebraic system. In particular, we develop the idea of applying the Mellin-Barnes integral representation of the monomial function for solving the extension problem and demonstrate how to achieve the same result following the fact that the solution to the universal trinomial system is polyhomogeneous. As a main result, we construct Puiseux expansions (centred at the origin) representing analytic continuations of the monomial function.

In the article we present a criterion for convergence of the Mellin-Barnes integral for zeros of a system of Laurent polynomials. Also we give a hypergeometric series for these zeros.