Some relations in the polynomial rings associated with polynomial mappings of the plane (part 1)

In this article, relations are obtained for some polynomials in n variables over the field of complex numbers. These polynomials are related to polynomial mapping of the plane with a constant Jacobian. As is known, the question of the invertibiliti of a polynomial mapping F with a constant Jacobian det J(F) was formulated by Keller O.H. in 1939. Based on the properties of these polynomials, a necessary and sufficient condition for the constancy of the det J(F) in the two - dimensional case is obtained. The second part of the article is devoted to the proof of the formulated theorem.