Generalization of Gelfond’s lemma on small values of integer polynomials to the multidimensional case

The paper establishes a relationship between the values of two integer polynomials without common roots on disjoint intervals of fixed length with the main characteristics of the polynomials – degree and height. The proved theorem can be considered as a two-dimensional generalization of Gelfond's lemma from the theory of transcendental numbers. The theorem can be used to estimate from above the Hausdorff dimension of a set of vectors that are approximated by conjugate algebraic numbers in a given order.