0. Introduction
1 Notation, basic definitions and two-dimensional case ......... 6
1.1 Notation ................................................ 7
1.2 Examples of polynomial automorphisms .................... 7
1.3 Degree, bidegree and multidegree ........................ 8
1.4 Jung and van der Kulk result ............................ 9
2 Main tools .................................................. 11
2.1 Poisson bracket and degree of polynomials .............. 11
2.2 Degree of a Poisson bracket and a linear change of
coordinates ............................................ 13
2.3 Shestakov-Umirbaev reductions .......................... 15
2.4 Some number theory ..................................... 18
3 Some useful results ......................................... 19
3.1 Some simple remarks .................................... 19
3.2 Reducibility of type I and II .......................... 21
3.3 Reducibility of type HI ................................ 22
3.4 Reducibility of type IV and Kuroda's result ............ 25
3.5 Reducibility and linear change of coordinates .......... 25
3.6 Relationship between the degree of the Poisson
bracket and the number of variables .................... 26
4 The case (p1, p2, p3) and its generalization ................ 28
4.1 The case (p1, p2, p3) .................................. 28
4.2 Some consequences ...................................... 29
4.3 Generalization ......................................... 30
4.4 The set mdeg(Aut( 3)) \ mdeg(Tame(C3)) ................. 30
5 The case (3, p2, p3) ........................................ 32
6 The case (4, p2, p3) ........................................ 33
6.1 The case (4, even, even) ............................... 33
6.2 The case (4, odd, odd) ................................. 33
6.3 The case (4, even, odd) ................................ 35
6.4 The case (4, odd, even) ................................ 37
7 The cases (p, p2, p3)) and (5, p2, p3) ...................... 39
7.1 The general case ....................................... 39
7.2 Tame automorphism of 3 with multidegree equal
(5, 6, 9) and the Jacobian Conjecture .................. 41
7.3 The case (p, 2(p - 2), 3(p - 2)) ....................... 43
8 Finiteness results .......................................... 44
9 Multidegree of the inverse of a polynomial automorphism of
2 .......................................................... 46
9.1 Multidegree and length of automorphisms of 2 .......... 46
9.2 The case of length 1 ................................... 49
9.3 The case (p2, p3)) ..................................... 50
9.4 The case (d,d) ......................................... 52
References ..................................................... 53
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