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An upper estimate for the norm of some positive integer solution of an arbitrary system of linear homogeneous equations with integer coefficients with nonempty set of positive integer solutions is obtained. This estimate is used to estimate from above the norm of a vector in which the components are powers of binomialswhose product is a difference polynomial of minimal possibleweight. The last result can be applied in the theory of beam steerable antenna arrays

Positive integer solutions of systems of linear equations and polynomials of small weight divisible by (1 − x) r