The Jacobian conjecture and the extensions of polynomial embeddings

Abstract

Let X be an algebraic subset of C". We say that X has the Abhyanka r -Moh property (AMP) if for any polynomial embedding 4~" X -~ OF" there exists a polynomial au tomorphism ~b of C" such that ~b is a restriction of this au tomorphism to the set X. In [3, 9] it was shown that if dim X is sufficiently small relatively to n, and if X has "nice" singularities then X has the AM P and the problem of extending the polynomial embedding of X into C" has (in general) many solutions. Unfortunately if dim X is big enough in compar ison with n the situation is more difficult. In particular, if X is a generic hypersurface of degree > n in ~'" (n > 2) then the problem of extending the polynomial embedding of X into C" has no more than one solution (see [5]). Abhyankar and Mob proved (see [1]) that a line in C 2 has the AMP. It can be easily inferred from this that the c r o s s K 2 = {X G ( ~ 2 : xl "x2 = 0} has the AMP. The analogous questions for a hyperplane and the n-cross K, = { x ~ C " : x l " . . . ' x , = 0} in the n-space (n > 3) are open (cf. [1] and Conjecture 12 in [6]). Moreover in the case n > 3 no example of a hypersurface which has the A M P is known. On the other hand, in [4] we showed that for every n > 2 the smooth hypersurface X,, = {x ~ C": Xl " . . . " x, = 2} (2 is any non-zero complex constant) does not have the AMP. We recall an important conjecture connected with this subject: