# Automorphismes et variables de l'anneau de polynômes A[y_1,...,y_n]

Abstract : In a polynomial ring in $n$ variables $A\n=A[y_1\tr y_n]$ with coefficients in a commutative ring $A$, a polynomial $p=p(y_1\tr y_n)$ is called a variable or an $A$-variable if there exists an ($A$-)automorphism $\alpha$ of $A\n$ such that $p=\alpha(y_1)$. In this thesis we give a quite general construction of variables of $A\n$ conjugating automorphisms of $A\n$ with automorphisms of $(\Quot A)\n$. We define residual variables which refer to polynomials that are variables modulo $\Max$ for every maximal ideal $\Max$ of $A$; in particular, when $A=\C\x=\C[x_1\tr x_k]$, one says $\xb$-residual variables. Of course, variables are residual variables, but what about the inverse implication? Using a Daigle and Freudenburg's result, we show that $\xb$-residual variables of $\C\x[y,z]$ are actually $\xb$-variables. Variables naturally appear in Abhyankar-Sathaye's embedding problem. A polynomial $p$ in $A\n$ is called an ($A$-)hyperplane if the quotient of $A\n$ by the principal ideal $(p)$ generated by $p$ is isomorphic to $A^{[n-1]}$. Variables are hyperplanes and the problem consists in studying the inverse implication. In an article, written jointly with M. Kaliman and M. Zaidenberg and which is the last part of this thesis, we study hyperplanes of $\C[x,y,z,u]$ of the form $p=f(x,y)u+g(x,y,z)$. Up to a change of the variables $x$ and $y$ we show that those hyperplanes are also $x$-residual variables. Moreover we prove that they are $x$-planes of $\Cx[y,z,u]$ and, that there exists an automorphism $\alpha$ of $A[y,z,u,v]$ such that $\alpha((p,v))=(y,v)$. In some cases, for example when $g$ has degree one in $z$, we manage to prove that they are $x$-variables. We also give a generalization of Wright's theorem showing that an $x$-plane of the form $f(x,y,z)u^n+g(x,y,z)$, where $n\geq 2$, is an $x$-variable. However the problem remains unsolved when we consider, for instance, the polynomial $y+x[xz+y(yu+z^2)]$ which, although being an $x$-plane and an $x$-residual variable, does not seem to be an $x$-variable.
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Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00001846
Contributor : Stéphane Vénéreau <>
Submitted on : Friday, October 18, 2002 - 5:07:09 PM
Last modification on : Wednesday, November 4, 2020 - 1:57:53 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 6:10:21 PM

### Identifiers

• HAL Id : tel-00001846, version 1

### Citation

Stéphane Vénéreau. Automorphismes et variables de l'anneau de polynômes A[y_1,...,y_n]. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2001. Français. ⟨tel-00001846⟩

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