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Образование Крыму » Информатика. Компьютеры » Polynomial automorphisms and the jacobian conjecture - Arno van den Essen

Polynomial automorphisms and the jacobian conjecture - Arno van den Essen

Название: Polynomial automorphisms and the jacobian conjecture
Автор: Arno van den Essen
Категория: Информатика. Компьютеры
Тип: Книга
Дата: 23.02.2009 11:06:03
Скачано: 66
Описание: It was in December 1985, when I first heard about the Jacobian Conjecture. I was in Paris to give a series of lectures on Algebraic Microlocalization at the Institute Henri Poincare. At that time my main interest was the study of Э-modules with regular singularities using algebraic methods. In my audience was a mathematician from Togo, his name Kossivi Adjamagbo (Pascal). He was working on a thesis concerning non-commutative determinants extending earlier work of Dieudonne from 1943. The main point of his interest in my lectures was that, using algebraic microlocalization, it is possible to embed a large class of non-commutative domains in a skew-field (this is not always possible for arbitrary non-commutative domains). After my first lecture he came to my office in Paris VI and invited me and my wife Sandra for dinner. During this dinner he explained to us that he had been working for three days and nights (without interruption!) on a famous conjecture in algebraic geometry called the Jacobian Conjecture and that he was close to a solution (this would happen many times later; at almost every occasion we met he came up with a new "solution"). However it was in June 1986, during another visit to Paris, that he explained the Jacobian Conjecture to me in more detail. That's how it all started for me. The Jacobian Conjecture originated from a paper [179] by Ott-Heinrich Keller in 1939 entitled "Ganze Cremona-Transformationen". In this paper he formulated the following question, now known as "Keller's problem": "given polynomials F\,..., F„ e 1\X\, ..., Xn] such that det JF = 1, where J F denotes the Jacobian matrix (jjf- J does it follow that each Xj can be expressed as a polynomial in F\,..,, Fn with coefficients in Ъ?". In the introduction of his paper Keller motivates his question as follows: "Fiir manche Integritdtsbereiche lasst sich eine lineare Basis A = («i,..., an) angeben, undjedes Element des Integritats-bereiches lasst sich als Linearkombination J2 А,я; mit ganzzahligen A,- darstellen. Durcheine unimodulare Lineartransformation erhal-ten wir eine neue Basis, und umgekehrt hdngtjede andere Basis В mit A vermo'ge einer unimodularen Lineartransformation zusammen. Wir haben also eine vollstdndige Ubersicht Uber die moglichen Basissystemen. Nun gibt es Integritdtsbereiche, die keine lineare Basis, wohl aber ein Basis {x\, X2.....xn) besitzen, mit deren Hilfe sich jedes Element als Polynom £ A,-,.....,„ •t'l' .. .x'," mit ganzzahligen A<,.....,-„ darstellen lusst. Ein Beispiel sind die sym- metrische Funktionen; fur sie sind die symmetrischen Grundfunktionen eine solche Basis. Es ergibt sich die Aufgabe, auch hier eine Obersicht alle moglichen Basissys-teme und die vermittelnden Transformationen zu gewinnen". "For many integral domains there exists a linear basis A = (a\,..., an) such that every element of this integral domain can be written as a linear combination £] A,a,-with integral coefficients. By a unimodular linear transformation we get a new basis and conversely every other basis В is related to A by means of a unimodular linear transformation. So we have a complete overview of all possible basis systems.
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