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Let k((x,y)) be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k((x,y)). We prove that if Ker(d) 6= k then Ker(d) =Ker( ), where is a jacobian derivation of k((x,y)). Moreover, we prove that Ker(d) is of the form k((h)), for some h 2 k((x,y)). 1. Introduction. Let k(x,y) be the ring of polynomials in two variables over a field k of characteristic zero. Let d be a nonzero derivation of k(x,y) and let A be its ring of constants. It is well known (see for example (4)) that if A 6= k then A = Ker( ), where is a jacobian derivation of k(x,y). It is also well known ((5) or (4)) that A is of the form k(h) for some h 2 k(x,y). In 1975, A. P loski ((6)) proved similar facts for derivations in convergent power series ring in two variables. In this paper we show that the above facts are also true for derivations in formal power series ring in two variables. 2. Preliminaries. If F is a nonzero power series from k((x,y)) then we denote by !(F ) the lowest homogeneous form of F , and we denote by o(F ) the order of F , that is, o(F ) = deg!(F ). Moreover, we assume that !(0) = 0 and o(0) = 1. It is clear that o(FG) = o(F ) + o(G) for all F,G 2 k((x,y)).

On the ring of constants for derivations of power series rings in two variables