A Generalization of Vinogradov's Mean Value Theorem

We obtain new upper bounds for the number of integral solutionsof a complete system of symmetric equations, which may be viewedas a multi-dimensional version of the system considered in Vinogradov'smean value theorem. We then use these bounds to obtain Weyl-typeestimates for an associated exponential sum in several variables.Finally, we apply the Hardy–Littlewood method to obtainasymptotic formulas for the number of solutions of the Vinogradov-typesystem and also of a related system connected to the problemof finding linear spaces on hypersurfaces. 2000 MathematicsSubject Classification 11D45, 11D72, 11L07, 11P55.