عنوان

Some aspects of the Jacobian conjecture :

Number

TLets750989

Title Proper

Some aspects of the Jacobian conjecture :

General Material Designation

[Thesis]

First Statement of Responsibility

Ali, A. Hamid A. Hussain

Title Proper by Another Author

the geometry of automorphisms of C2

Subsequent Statement of Responsibility

O'Connor, John Joseph

Name of Publisher, Distributor, etc.

University of St Andrews

Date of Publication, Distribution, etc.

1987

Dissertation or thesis details and type of degree

Thesis (Ph.D.)

Text preceding or following the note

1987

Text of Note

We consider the affine varieties which arise by considering invertible polynomial maps from C2 to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.

QA404.5A6 ; Orthogonal polynomials

Ali, A. Hamid A. Hussain

O'Connor, John Joseph

University of St Andrews

Electronic name

p

[Thesis]

276903

a

Y