Examples of computing power sums of roots of systems of equations

Examples of computing power sums of roots of systems of equations, including transcendental, are considered. Since the number of roots of such systems is, as a rule, infinite, it is necessary to study the power sums of roots in a negative degree. Formulas for finding residue integrals, their connection with power sums of roots to a negative degree, multidimensional analogues of Waring's formulas are given.