Poisson Brackets and Two-generated Subalgebras of Rings of Polynomials

Let A = F [x1, x2, . . . , xn] be a ring of polynomials over a field F on the variables x1, x2, . . . , xn. It is well known (see, for example, [11]) that the study of automorphisms of the algebra A is closely related with the description of its subalgebras. By the theorem of P. M. Cohn [4], a subalgebra of the algebra F [x] is free if and only if it is integrally closed. The theorem of A. Zaks [13] says that the Dedekind subalgebras of the algebra A are rings of polynomials in a single variable. A. Nowicki and M. Nagata [8] proved that the kernel of any nontrivial derivation of the algebra F [x, y], char(F ) = 0, is also a ring of polynomials in a single generator. An original solution of the occurrence problem for the algebra A, using the Groebner basis, was given by D. Shannon and M. Sweedler [9]. However, the method of the Groebner basis does not give any information about the structure of concrete subalgebras. Recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by G. Noskov [7]. The present paper is devoted to the investigation of the structure of twogenerated subalgebras of A. In the sequel, we always assume that F is an arbitrary field of characteristic 0. Let us denote by f̄ the highest homogeneous part of an element f ∈ A, and by 〈f1, f2, . . . , fk〉 the subalgebra of A generated by the elements f1, f2, . . . , fk ∈ A. Definition 1. A pair of polynomials f1, f2 ∈ A is called ∗-reduced if they satisfy the following conditions: 1) f̄1, f̄2 are algebraically dependent; 2) f1, f2 are algebraically independent; 3) f̄1 / ∈ 〈f̄2〉, f̄2 / ∈ 〈f̄1〉. Recall that a pair f1, f2 with condition 3) is usually called reduced. Condition 1) means that we exclude the trivial case when f̄1, f̄2 are algebraically independent. We do not consider the case when f1, f2 are algebraically dependent. Concerning this case, recall the well-known theorem of S. S. Abhyankar and T. -T. Moh [1], which says that if f, g ∈ F [x] and 〈f, g〉 = F [x], then f̄ ∈ 〈ḡ〉 or ḡ ∈ 〈f̄〉.