On the convergence exponent of trigonometric integrals

The method of trigonometric sums is one of the most powerful tools in analytic number theory. In particular trigonometric integrals play an important role. Moreover, many problems in both mathematical physics and theory of probability lead to investigation of trigonometric integrals. Namely, it is important asymptotic behavior, estimation and summation exponent with respect to parameters of such integrals.
In this paper we consider multi-dimensional trigonometric integrals with polynomial phases and get a lower estimation for convergence exponent of functions defined by such integrals. Moreover, we obtain explicit value of convergence exponent in particular cases.