On solutions and Waring’s formulas for systems of $n$ algebraic equations for $n$ unknowns
Vladimir R. Kulikov, V. A. Stepanenko · St Petersburg Mathematical Journal · 2015
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.