Multidegrees of tame automorphisms of Cn

Let F = (F-1, ... , F-n) : C-n -> C-n be a polynomial mapping. By the multidegree of F we mean mdegF = (deg F-1, ... , deg F-n) is an element of N-n. The aim of this paper is to study the following problem (especially for n = 3): for which sequence (d(1), ... , d(n)) is an element of N-n is there a tame automorphism, F of C-n such that mdeg F = (d(1), ... , d(n))? In other words we investigate the set mdeg(Tame(C-n)), where Tame(C-n) denotes the group of tame automorphisms of C-n. Since mdeg(Tame(C-n)) is invariant under permutations of coordinates, we may focus on the set {((d(1), ... , d(n)) : d(1) <= ... <= d(n)} boolean AND mdeg(Tame(C-n)). Obviously, we have {(1, d(2), d(3)) : 1 <= d(2) <= d(3)} boolean AND mdeg(Tame(C-3)) = {(1, d(2), d(3)) : 1 <= d(2) <= d(3)}. Not obvious, but still easy to prove is the equality mdeg(Tame(C-3)) boolean AND {(2, d(2), d(3)) : 2 <= d(2) <= d(3)} = {(2, d(2), d(3)) :2 <= d(2) <= d(3)}. We give a complete description of the sets {(3, d(2), d(3)) : 3 <= d(2) <= d(3)} boolean AND mdeg(Tame(C-3)) and {(5, d(2), d(3)) : 5 <= d(2) <= d(3)} boolean AND mdeg(Tame(C-3)). In the examination of the last set the most difficult part is to prove that (5, 6, 9) is not an element of mdeg(Tame(C-3)). To do this, we use the two-dimensional Jacobian Conjecture (which is true for low degrees) and the Jung van der Kuljc Theorem. As a surprising consequence of the method used in proving that (5, 6, 9) is not an element of mdeg(Tame(C-3)), we show that the existence of a tame automorphism F of C-3 with mdeg F = (37, 70,105) implies that the two-dimensional Jacobian Conjecture is not true. Also, we give a complete description of the following sets: {(p(1), p(2), d(3)) : 2 < p(1) < p(2) <= d(3), p(1), p(2) prime numbers} boolean AND mdeg(Tame(C-3)), {(d(1), d(2), d(3)) : d(1) <= d(2) <= d(3), d(1), d(2) is an element of 2N + 1, gcd(d(1), d(2)) = 1} boolean AND mdeg(Tarne(C-3)). Using the description of the last set we show that mdeg(Aut(C-3)) \ mdeg(Tame(C-3)) is infinite. We also obtain a (still incomplete) description of the set mdeg(Tame(C-3)) boolean AND {(4, d(2), d(3)) : 4 <= d(2) <= d(3)} and we give complete information about mdeg F-1 for F is an element of Aut(C-2).