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This paper investigates the upper bound of the number of integer (natural)solutions of inhomogeneous algebraic Diophantine diagonal equations withinteger coefficients without a free member via the circle method of Hardy andLittlewood. Author found the upper bound of the number of natural solutions ofinhomogeneous algebraic Diophantine diagonal equations with explicit variable.He developed a method in the paper, which allows you to perform the lowestimate of the number of natural (integer) solutions of algebraic Diophantineequation with integer coefficients. Author obtained a lower estimate (with thismethod) of the number of integer (natural) solutions for certain kinds ofinhomogeneous algebraic Diophantine diagonal equations with integercoefficients with any number of variables (including Thue equation).

Estimating of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients