Let F = (F₁,...,Fₙ): ℂⁿ → ℂⁿ be a polynomial mapping. By the multidegree of F we mean mdeg F = (deg F₁, ..., deg Fₙ) ∈ ℕ ⁿ. The aim of this paper is to study the following problem (especially for n = 3): for which sequence (d₁,...,dₙ) ∈ ℕ ⁿ is there a tame automorphism F of ℂⁿ such that mdeg F = (d₁,..., dₙ)? In other words we investigate the set mdeg(Tame(ℂⁿ)), where Tame(ℂⁿ) denotes the group of tame automorphisms of ℂⁿ.
Since mdeg(Tame(ℂⁿ)) is invariant under permutations of coordinates, we may focus on the set {(d₁,...,dₙ): d₁ ≤ ⋯ ≤ dₙ} ∩ mdeg (Tame(ℂⁿ)).
Obviously, we have {(1,d₂,d₃): 1 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)) = {(1,d₂,d₃): 1 ≤ d₂ ≤ d₃}. Not obvious, but still easy to prove is the equality mdeg(Tame(ℂ³)) ∩ {(2,d₂,d₃): 2 ≤ d₂ ≤ d₃} = {(2,d₂,d₃): 2 ≤ d₂ ≤ d₃}.
We give a complete description of the sets {(3,d₂,d₃): 3 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)) and {(5,d₂,d₃): 5 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)). In the examination of the last set the most difficult part is to prove that (5,6,9) ∉ mdeg(Tame(ℂ³)). To do this, we use the two-dimensional Jacobian Conjecture (which is true for low degrees) and the Jung-van der Kulk Theorem.
As a surprising consequence of the method used in proving that (5,6,9) ∉ mdeg(Tame(ℂ³)), we show that the existence of a tame automorphism F of ℂ³ 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: {(p₁,p₂,d₃): 2 < p₁ < p₂ ≤ d₃, p₁,p₂ prime numbers} ∩ mdeg(Tame(ℂ³)), {(d₁,d₂,d₃): d₁ ≤ d₂ ≤ d₃, d₁,d₂ ∈ 2ℕ +1, gcd(d₁,d₂) = 1} ∩ mdeg(Tame(ℂ³)). Using the description of the last set we show that mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) is infinite.
We also obtain a (still incomplete) description of the set mdeg(Tame(ℂ³)) ∩ {(4,d₂,d₃): 4 ≤ d₂ ≤ d₃} and we give complete information about $mdeg F^{-1}$ for F ∈ Aut(ℂ²).