  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.
