On the exponents of the convergence of singular integrals and singular series of a multivariate problem

Abstract

In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I. M. Vinogradov's method. Here we obtain for $$n=r=2$$ lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for $$P\to\infty$$ for the number of solutions of the following system of Diophantine equations
$$
\sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}\dots x_{r,j}^{t_r}=0,\quad 0\leq t_1,\dots, t_r\leq n,
$$
where $$n\geq 2,r\geq 1, k$$ are natural numbers, moreover an each variable $$x_{i,j}$$ can take
all integer values from 1 to $$P\geq 1.$$