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Stefan Maubach

Publications and source records attributed to Stefan Maubach.

At least 19 recordsLinked to original sources
arXiv2017

On Maximal Subalgebras

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, t^{-1}, y]$. To the authors' knowledge, this is the first such classification result for an algebra of dimension $> 1$. In the course of this study, we classify also all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that contain a coordinate. Furthermore, we give examples of maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that do not contain a coordinate.

math.AC↗
arXiv2015

A new formulation of the Jacobian Conjecture in characteristic $p$

The Jacobian Conjecture uses the equation $det(Jac(F))\in k^*$, which is a very short way to write down many equations putting restrictions on the coefficients of a polynomial map $F$. In characteristic $p$ these equations do not suffice to (conjecturally) force a polynomial map to be invertible. In this article, we describe how to construct the conjecturally sufficient equations in characteristic $p$ forcing a polynomial map to be invertible. This provides an (alternative to Adjamagbo's formulation) definition of the Jacobian Conjecture in characteristic $p$. We strengthen this formulation by investigating some special cases and by linking it to the regular Jacobian Conjecture in characteristic zero.

math.AC↗
arXiv2014

The profinite polynomial automorphism group

We introduce an extension of the (tame) polynomial automorphism group over finite fields: the profinite (tame) polynomial automorphism group, which is obtained by putting a natural topology on the automorphism group. We show that most known candidate non-tame automorphisms are inside the profinite tame polynomial automorphism group, giving another result showing that tame maps are potentially "dense" inside the set of automorphisms. We study the profinite tame automorphism group and show that it is not far from the set of bijections obtained by endomorphisms.

math.AG↗
arXiv2013

Invariants and conjugacy classes of triangular polynomial maps

In this article, we classify invariants and conjugacy classes of triangular polynomial maps. We make these classifications in dimension 2 over domains containing $\Q$, dimension 2 over fields of characteristic $p$, and dimension 3 over fields of characteristic zero. We discuss the generic characteristic 0 case. We determine the invariants and conjugacy classes of strictly triangular maps of maximal order in all dimensions over fields of characteristic $p$. They turn out to be equivalent to a map of the form $(x_1+f_1,\ldots,x_n+f_n)$ where $f_i\in x_n^{p-1}k[x_{i+1}^p,\ldots,x_n^p]$ if $1\leq i\leq n-1$ and $f_n\in k^*$.

math.AG↗
arXiv2013

Triangular polynomial maps in characteristic $p$

This paper came to existence out of the desire to understand iterations of strictly triangular polynomial maps over finite fields. This resulted in two connected results: First, we give a generalization of $\F_p$-actions on $\F_p^n$ and their description by "locally iterative higher derivations", namely $\Z$-actions on $\F_p^n$ and show how to describe them by what we call "$\Z$-flows". We prove equivalence between locally finite polynomial automorphisms (LFPEs) over finite fields and $\Z$-flows over finite fields. We elaborate on $\Z$-flows of strictly triangular polynomial maps. Second, we describe how one can efficiently evaluate iterations of triangular polynomial permutations on $\F_p^n$ which have only one orbit. We do this by determining the equivalence classes in the triangular permutation group of such elements. We show how to conjugate them all to the map $z\lp z+1$ on the ring $\Z/p^n\Z$ (which is identified with $\F_{p^n}$), making iterations trivial. An application in the form of fast-forward functions from cryptography is given.

math.AG↗
arXiv2011

Polynomial endomorphisms over finite fields: experimental results

Given a finite field $\F_q$ and $n\in \N^*$, one could try to compute all polynomial endomorphisms $\F_q^n\lp \F_q^n$ up to a certain degree with a specific property. We consider the case $n=3$. If the degree is low (like 2,3, or 4) and the finite field is small ($q\leq 7$) then some of the computations are still feasible. In this article we study the following properties of endomorphisms: being a bijection of $\F_q^n\lp \F_q^n$, being a polynomial automorphism, being a {\em Mock automorphism}, and being a locally finite polynomial automorphism. In the resulting tables, we point out a few interesting objects, and pose some interesting conjectures which surfaced through our computations.

math.AG↗
arXiv2010

Locally tame plane polynomial automorphisms

For automorphisms of a polynomial ring in two variables over a domain R, we show that local tameness implies global tameness provided that every 2-generated invertible R-module is free. We give many examples illustrating this property.

math.AG↗
arXiv2010

Rigid rings and Makar-Limanov techniques

A ring is rigid if there is no nonzero locally nilpotent derivation on it. In terms of algebraic geometry, a rigid coordinate ring corresponds to an algebraic affine variety which does not allow any nontrivial algebraic additive group action. Even though it is thought that "generic" rings are rigid, it is far from trivial to show that a ring is rigid. In this paper we provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call "parametrization techniques" and "filtration techniques". We provide many little tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple hypersurfaces for which the question of rigidity remains unsettled.

math.AG↗
arXiv2010

Computing preimages of points and curves under polynomial maps

In this paper, we give two algorithms to compute preimages of curves under polynomial endomorphisms. In particular, this gives an efficient way of computing preimages of points. Furthermore, we explain the abstract setting under which one can iteratively compute the inverse of a polynomial automorphism.

math.AG↗
arXiv2010

Unipotent group actions on affine varieties

Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial, $O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that $O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is $U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye conjecture and a new equivalent formulation of the Sathaye conjecture are made.

math.AG↗
arXiv2009

Polynomial automorphisms over finite fields: Mimicking non-tame and tame maps by the Derksen group

If $F$ is a polynomial automorphism over a finite field $\F_q$ in dimension $n$, then it induces a bijection $π_{q^r}(F)$ of $(\F_{q^r})^n$ for every $r\in \N^*$. We say that $F$ can be `mimicked' by elements of a certain group of automorphisms $\mathcal{G}$ if there are $g_r\in \mathcal{G}$ such that $π_{q^r}(g_r)=π_{q^r}(F)$. We show that the Nagata automorphism (and any other automorphism in three variables fixing one variable) can be mimicked by tame automorphisms. This on the one hand removes the hope of showing that such an automorphism is non-tame by studying one bijection it induces, but on the other hand indicates that the bijections of tame automorphisms coincide with bijections of all automorphisms. In section 5 we show that the whole tame group $\TA_n(\F_q)$ in turn can be mimicked by the Derksen subgroup $\DA_n(\F_q)$, which is the subgroup generated by affine maps and one particular element. It is known that if a field $k$ has characteristic zero, then $\TA_n(k)=\DA_n(k)$, which is not expected to be true for characteristic $p$. In section 6 we consider the subgroups $\GLIN_n(k)$ and $\GTAM_n(k)$ of the polynomial automorphism group (which are candidates to equal the entire automorphism group). We show that $\GLIN_n(\F_q)=\GTAM_n(\F_q)$ if $q\not = 2$, and $\GLIN_n(\F_2)\subsetneq \GTAM_n(\F_2)$.

math.AG↗
arXiv2008

A note on k[z]-automorphisms in two variables

We prove that for a polynomial $f\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x,y,z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method - essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$.

math.AC↗
arXiv2008

A commuting derivations theorem on UFDs

Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-action on $k^{n+1}$ must have invariant ring $k[f]$ where $f$ is a coordinate. In this paper we prove a statement (theorem \ref{CDH2}) where we assume less on $A$ ($A$ is a {\sc UFD} over $k$ of transcendence degree $n+1$ satisfying $A^*=k$) and prove less ($A/(f-α)$ is a polynomial ring for all but finitely many $α$). Under certain additional conditions (the $D_i$ are linearly independent modulo $(f-α)$ for each $α\in k$) we prove that $A$ is a polynomial ring itself and $f$ is a coordinate. This statement is proven even more generally by replacing ``free unipotent action of dimension $n$'' for ``$G_a^n$-action''. We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture.

math.AG↗
arXiv2008

The Nagata automorphism is shifted linearizable

A polynomial automorphism $F$ is called {\em shifted linearizable} if there exists a linear map $L$ such that $LF$ is linearizable. We prove that the Nagata automorphism $N:=(X-YΔ-ZΔ^2,Y+ZΔ, Z)$ where $Δ=XZ+Y^2$ is shifted linearizable. More precisely, defining $L_{(a,b,c)}$ as the diagonal linear map having $a,b,c$ on its diagonal, we prove that if $ac=b^2$, then $L_{(a,b,c)}N$ is linearizable if and only if $bc\not = 1$. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question.

math.AG↗
arXiv2008

A Characterization of Semisimple Plane Polynomial Automorphisms

It is well-known that an element of the linear group ${\rm GL}_n(\C)$ is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.

math.AG↗
arXiv2008

A problem on polynomial maps over finite fields

This is a short note that explains a problem on polynomial maps over finite fields for non-experts. The problem is: Do there exist odd polynomial automorphisms over the finite fields with 4,8,16,32,64,... elements? The explanation is very, very basic. Also, the background of the problem is given, and why it is of such importance. This all with the idea that the problem enters the world of discrete mathematics, and can be approached from completely different angles than normally used by people working in Affine Algebraic Geometry.

math.CO↗
arXiv2007

\C-flows A^z of linear maps A expressed in terms of A^{-1},A^{-2},...,A^{-n} and analytic functions of z

Suppose A\in GL_n(\C) has a relation A^p=c_{p-1}A^{p-1}+.... + c_1 A+ c_0I where the c_i in \C. This article describes how to construct analytic functions c_i(z) such that A^z=c_{p-1}(z)A^{p-1}+... + c_1(z) A+ c_0(z)I . One of the theorems gives a possible description of the c_i(z): c_i(z)=C^zαwhere C\in Mat_p(\C) is (similar to) the companion matrix of X^p-c_{p-1}X^{p-1}-... -c_1X-c_0I, and α:= (c_{p-1},...,c_1,c_0)^t.

math.AC↗