A refinement of the estimates of the arithmetic minimum of the product of nonhomogeneous linear forms for large dimensions

Let M _n be the nonhomogeneous arithmetic minimum of the lattice Λ ? ?ⁿ of determinant Δ>0. It is proved that forn?1,1·n⁵, we have . One obtains other estimates of a similar type in the nonhomogeneous Minkowski conjecture, obtained for the first time by N. G. Chebotarev. These estimates strengthen the results of B. P. Skubenko.