Abstract
Let denote the smallest for which the system of equations (1)is solvable in nonnegative integers for all sufficiently large natural numbers which satisfy the following conditions: 1) The singular integral of the system (1) satisfies the inequality (the order conditions). 2) The system of equations () is solvable in integers (the arithmetic conditions).In 1937, K. K. Mardzhanishvili proved that . G. I. Arkhipov has recently obtained upper and lower estimates for having the same order of magnitude: ().In this paper, the upper estimate for is reduced to (2)in particular, the asymptotic formula is obtained. It is conjectured that the estimate (2) is best possible.Bibliography: 20 titles.
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