Improved bounds for the total variation distance between stochastic polynomials

The paper studies upper bounds for the total variation distance between two polynomials of a special form in random vectors satisfying the Doeblin-type condition. Our approach is based on the recent results concerning Nikolskii–Besov-type smoothness of distribution densities of polynomials in logarithmically concave random vectors. The main results of the paper improve previously obtained estimates of Nourdin–Poly and Bally–Caramellino.