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We show that certain compact polynomially convex subsets of C' remain polynomially convex under sufficiently small C2 perturbations. 1. Statement of the results. Let M be a Stein manifold. Denote by (9(M) the algebra of all holomorphic functions on M with the standard topology of uniform convergence on compact subsets. A compact subset K of M is said to be (9(M)-convex if for every point x E M \ K there is a holomorphic function f E (9(M) such that If(x) I> sup If(y) 1 yE K Since the holomorphic polynomials are dense in the algebra (9(C n) of holomorphic functions on Cn , an (9(Cn)-convex subset of CCn is just a polynomially convex subset. Given a compact (9(M)-convex subset K of M, an open neighborhood U of K and a Ck diffeomorphism ' of U onto an open subset I(U) in M, we ask whether the set I(K) is also (9(M)-convex provided that ' is sufficiently close to the identity on U in the Ck sense. In other words, is (9(M)-convexity a stable property under smooth perturbations? In general this is not so as the following example shows. EXAMPLE 1. Let M = C2 and K= {(z,O) E C2: Izi 1}. Clearly K is convex and hence polynomially convex. The diffeomorphisms '': C2 C2 given by 'I'(z,W) = (Z,W + El ) E >? 0, are close to the identity in the C' sense for small E, but the set *e(K) = {(z,lzl2): |zL 1} is not polynomially convex for any E > 0 since it contains the boundary of the analytic disk Av6 6 = {(Z,E82eio): Izi< I8} for each 8 E [0,1] and G Eli R. These disks fill an open subset of C2 that is contained in the polynomial hull of ''(K) according to the maximum principle. Received by the editors February 8, 1985 and, in revised form, April 22, 1985. 1980 Mathematics Subject Classification. Primary 32E20. 'Research supported in part by a Sloan Foundation Predoctoral Fellowship. ?c1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Stability of polynomial convexity of totally real sets