On the Arkhipov–Karatsuba multivariate system of congruences

Abstract

The Arkhipov–Karatsuba multivariate system of congruences modulo any prime greater than the degrees of forms in this system is solvable for any right-hand sides and any number of variables larger than 8(n + 1)mlog₂(rn) + 12(n + 1)m + 4(n + 1), where n is the degree of the forms in the system and \(m = \left( {\begin{array}{*{20}{c}} {n + r - 1} \\ {r - 1} \end{array}} \right)\) is the number of congruences.