Alexei Belov-Kanel, Andrey Elishev, Jie-Tai Yu
In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every diagonal matrix preserves, up to composition with a linear inner automorphism, every tame automorphism.
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It is proved that every z-automorphism (z-coordinate, respectively) of the free associative algebra F<x,y,z> over an arbitrary field F is stably tame.
In this paper, we discuss subgroups of the automorphism group of the polynomial ring in n variables over a field of characteristic zero. An automorphism F is said to be co-tame if the subgroup generated by F and affine automorphisms contains the tame subgroup. In 2017, Edo-Lewis gave a sufficient condition for co-tameness of automorphisms. Let EL(n) be the set of all automorphisms satisfying Edo-Lewis's condition. Then, for a certain topology on the automorphism group of the polynomial ring, any element of the closure of EL(n) is co-tame. Moreover, all the co-tame automorphisms previously know…
We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of $\C^3$ is not closed, when the latter is seen as an infinite dimensional algebraic group.tomorphism group of affine three-space
We construct explicitly a family of proper subgroups of the tame automorphism group of affine three-space (in any characteristic) which are generated by the affine subgroup and a non-affine tame automorphism. One important corollary is the titular result that settles negatively the open question (in characteristic zero) of whether the affine subgroup is a maximal subgroup of the tame automorphism group. We also prove that all groups of this family have the structure of an amalgamated free product of the affine group and a finite group over their intersection.
A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms, including $3$-triangular automorphisms, are co-tame. Of particular interest, if $n=3$, we show that the statement "Every $m$-triangular automorphism is either affine or co-tame" is true if and only if $m \leq 3$; this improves upon positive results of Bodnarchuk (for $m \leq 2$, in any dimension $n$) and negative results of the authors (for $m \geq 6$, $n=3$). The main technica…
An automorphism $F$ of the polynomial ring in $n$ variables over a field of characteristic zero is said to be {\it co-tame} if the subgroup of the automorphism group of the polynomial ring generated by $F$ and affine automorphisms contains the tame subgroup. There exist many examples of such an $F$, and several sufficient conditions for co-tameness are already known. In 2015, Edo-Lewis gave the first example of a non-cotame automorphism, which is a tame automorphism of the polynomial ring in three variables. In this paper, we give the first example of a non-cotame automorphism which is not tam…
Recently, Edo-Poloni constructed a family of tame automorphisms of a polynomial ring in three variables which degenerates to a wild automorphism. In this note, we generalize the example by a different method.
We say that a polynomial automorphism $ϕ$ in $n$ variables is stably co-tame if the tame subgroup in $n$ variables is contained in the subgroup generated by $ϕ$ and affine automorphisms in $n+1$ variables. In this paper, we give conditions for stably co-tameness of 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.
Let F=(F_1,...,F_n):C^n --> C^n be any polynomial mapping. By multidegree of F, denoted mdeg F, we call the sequence of positive integers (deg F_1,...,F_n). In this paper we addres the following problem: for which sequence (d_1,...,d_n) there is an automorphism or tame automorphism F:C^n --> C^n with mdeg F=(d_1,...,d_n}. We proved, among other things, that there is no tame automorphism F:C^3 --> C^3 with mdeg F=(3,4,5).
It is known that not each triple (d_1,d_2},d_3) of positive integers is a multidegree of a tame automorphism of C^3. In this paper we show that there is no tame automorphism of C^3 with multidegree (4,5,6). To do this we show that there is no pair of polynomials P,Q in C[x,y,z] with certain properties. These properties do not seem particularly restrictive so the non-existence result can be interesting in its own right.
In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.
The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field…
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.
In this note we prove that the sequence (3,d_2,d_3), where d_3>= d_2>= 3, is the multidegee of some tame automorphism of C^3, if and only if 3|d_2 or d_3 is a linaer combination of 3 and d_2 with coefficients in N.
We prove the strong Tits alternative for the 3-dimensional tame automorphism group over a field K of characteristic zero.
We prove the existence of wild automorphisms on an affine quadric threefold. The method we use is an adaptation of the one used by Shestakov and Umirbaev to prove the existence of wild automorphisms on the affine three dimensional space.
In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame aut…
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, b…
An explicit tame factorization of the four-variable Anick automorphism
本文给出四变量多项式环中Anick自同构的显式 tame 分解，含16个固定z的初等自同构，定义在ℤ[1/2]上，未涉及特征2情形。
We study the structure of length three polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. These results are used to prove that if $\text{SL}_m(R[X_1,X_2,..., X_n]) = \text{E}_m(R[X_1,X_2,..., X_n])$ for all $n,\ge 0$ and for all $m \ge 3$ then all length three polynomial automorphisms of $R[X,Y]$ are stably tame.
We describe a set of defining relations of the tame automorphism group TA_3(F) of the polynomial algebra F[x_1,x_2,x_3] in variables x_1,x_2,x_3 over an arbitrary field F of characteristic 0.
It is shown the the tame subgroup $\text{TA}_3(\mathbb C)$ of the group $\text{GA}_3(\mathbb C)$ of polynomials automorphisms of ${\mathbb C}^3$ can be realized as the product of three subgroups, amalgamated along pairwise intersections, in a manner that generalizes the well-known amalgamated free product structure of $\text{TA}_2(\mathbb C)$ (which coincides with $\text{GA}_2(\mathbb C)$ by Jung's Theorem). The result follows from defining relations for $\text{TA}_3(\mathbb C)$ given by U. U. Umirbaev.
In this note we show that the set mdeg(Aut(C^3)) mdeg(Tame(C^3)) is not empty. Moreover we show that this set has infinitely many elements. Since for the famous Nagata's example N of wild automorphism, mdeg N =(5,3,1) is an element of mdeg(Tame(C^3)) and since for other known examples of wild automorphisms the multidegree is of the form (1,d_2,d_3) (after permutation if neccesary), then we give the very first exmple of wild automorphism F of C^3 such that mdeg F does not belong to mdeg(Tame(C^3)). We also show that, if d_1,d_2 are odd numbers such that gcd (d_1,d_2) =1, then (d_1,d_2,d_3) belo…
We study the group Tame($\mathbf A^3$) of tame automorphisms of the 3-dimensional affine space, over a field of characteristic zero. We recover, in a unified and (hopefully) simplified way, previous results of Kuroda, Shestakov, Umirbaev and Wright, about the theory of reduction and the relations in Tame($\mathbf A^3$). The novelty in our presentation is the emphasis on a simply connected 2-dimensional simplicial complex on which Tame($\mathbf A^3$) acts by isometries.
Let d_3 >= p_2 > p_1 >= 3 be integers such that p_1,p_2 are prime numbers. In this paper we show that the sequence (p_1,p_2,d_3) is the multidegree of some tame automorphisms of C^3 if and only if d_3 is in p_1*N+p_2*N, i.e. if and only if d_3 is a linear combination of p_1 and p_2 with coefficients in N.
We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL(2,$\mathbb{C}$). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL$_2$) is linearizable, and that Tame(SL$_2$) satisfies the Tits alternative.
On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups
本文利用驯顺逼近研究多项式自同构 Ind-群的自同构，证明了 n≥3 时多项式代数自同构群的 Ind-概形自同构均为内自同构，并由此给出自同构群提升问题的否定解，进而探讨其与雅可比猜想的关系。
Let F=(F_1,...,F_n):C^n --> C^n be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F_1,...,deg F_n), an element of N^n. The aim of this paper is to study the following problem (especially for n=3): for which sequence (d_1,...,d_n) in N^n there is a tame automorphism F of C^n such that mdeg F=(d_1,...,d_n). In other words we investigate the set mdeg(Tame(C^n)), where Tame(C^n) denotes the group of tame automorphisms of C^n and mdeg denotes the mapping from the set of polynomial endomorphisms of C^n into the set N^n. Since for all permutation s of {1,...,n} we have …
Tame polynomial automorphisms
本文综述了仿射空间驯服多项式自同构群的研究，通过陪集复形与赋值复形上的作用，探讨维数2、3及保持非退化二次型的维数4情形下的正规子群、有限子群分类、Tits 交替与动力次数等问题。
Infinite transitivity of tame groups of automorphisms of affine spaces
该研究针对有限域上n维仿射空间的驯服自同构群，探究其对m个点有序子集的作用，构造可映射有序序列的驯服自同构并证明其复杂度的上下界，运用多数学分支方法获得相关定性与定量结果。
Degree growth for tame automorphisms of an affine quadric threefold
本文利用赋值估计和方复形上的群作用，证明了保持仿射二次三维簇的驯顺自同构的动力次数避开 1 和 4/3 之间的值，并刻画了有限多个此类自同构随机乘积的次数增长指数为正的条件。
In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted over the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which whole the automorphism group o…
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ of polynomial automorphisms of complex affine $n$-space $\mathbb{A}^n=\mathbb{C}^n$ is inner up to field automorphisms when restricted to the subgroup $T \mathcal{G}_n$ of tame automorphisms. This generalizes a result of \textsc{Julie Deserti} who proved this in dimension $n=2$ where all automorphisms are tame: $T \mathcal{G}_2 = \mathcal{G}_2$.
In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the "generalized Shestakov-Umirbaev inequality", which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we show that no tame automorphism of a polynomial ring admits a reduction of type IV.
Finite automorphism groups of hypersurfaces via Bott vanishing
本文通过Bott消失定理给出上同调判据，证明驯特征下光滑Deligne-Mumford栈中超曲面的自同构群概形离散，并应用于环面轨道簇，获得拟光滑加权射影超曲面及射影空间乘积中超曲面的有限性结果，同时指出方法失效的例外窗口。
On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]
本文研究了K[X,Y]局部有限导数的tame等距群，通过Van den Essen的分类确定各规范形式对应的tame等距群，并证明其与关联的指数自同构的tame等距群一致，与全自同构群行为不同。
We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial which is tame. As an application we show the existence of conjugated polynomials in a number field which are not topologically equivalent
We prove that, in characteristic zero, closed subgroups of the polynomial automorphisms group containing the affine group contain the whole tame group.
Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $γ$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $γ$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\mathcal{X})$ is a finite group, and unless $\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few ex…
We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or "wildness", of Nagata's automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism in any dimension and reprove the tameness of automorphisms in dimension two.
We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift depending on the unipotent or non-unipotent monodromy part.
Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $θ\colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $θ$ fixes a closed torus $T \subseteq \mathcal{G}$, which normalizes $\mathcal{U}$ and the action of $T$ on $\mathcal{U}$ by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona gr…
We analyse the Gauss-Manin system of differential equations---and its Fourier transform---attached to regular functions satisfying a tameness assupmption on a smooth affine variety over C (e.g. tame polynomials on C^{n+1}). We give a solution to the Birkhoff problem and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities.
It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K<x,y,z>$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF<x_1,...,x_n>}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF<x,y>)}$.
We prove that for every pair of positive integers a, b there exists a number c_0 such that for every c>=c_0 one can find a tame polynomial automorphism of C^3 with multidegree equal to (a,b,c).
An algorithm to compute a good basis of the Brieskorn lattice of a cohomologically tame polynomial is described. This algorithm is based on the results of C. Sabbah and generalizes the algorithm by A. Douai for convenient Newton non-degenerate polynomials.
The inversion formula is given for automorphisms of the Weyl algebras with polynomial coefficients over a field of characteristic zero. The theorem of Gabber on the degree of polynomial automorphism is extended. It is proved that any automorphism of the Weyl algebra with polynomial coefficients is determined by its left (right) faces (extending the result of J. H. McKay and S. S.-S. Wang for polynomial automorphisms).
The groups of automorphisms of the Lie algebras of unitriangular polynomial derivations are found.
The group of automorphisms is found for the Lie algebra of polynomial vector fields with constant divergence.
Iterated Cartier Flux and Non-Finite Generation of Tame Polynomial Automorphism Groups over Finite Fields
该研究针对有限域上的驯服多项式自同构群，通过迭代嘉当通量的性质证明其仅在一维时有限生成，进而证实了 $n\ge3$ 时的 Maubach–Willems 有限生成猜想。
Explicit generators are found for the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in $n$ variables. An analogue of the polynomial Jacobian homomorphism is found.
We prove that the group of automorphisms of the Lie algebra $\Der_K (P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,..., x_n]$ over a field of characteristic zero is canonically isomorphic to the the group of automorphisms of the polynomial algebra $P_n$.
We prove that every monomorphism of the Lie algebra $\ggu_n$ of unitriangular derivations of the polynomial algebra $P_n=K[x_1,..., x_n]$ is an automorphism.
Nagata's conjecture on a polynomial automorphism in positive characteristic
本文在特征至少为7的域上证明了Nagata关于三元多项式环特定自同构为野性的猜想，首次在正特征下确认野性自同构的存在。
We compare each coefficient of the reduced characteristic polynomial of a simple arrangement and that of its Ziegler restriction. As a consequence we can show that the former is not less than the latter in the category of tame arrangements. This is a generalization of Yoshinaga's freeness criterion for 3-arrangements and also the recent result by the author and Yoshinaga. As a corollary, we can prove that a free arrangement is a minimal chamber arrangement, and we can give a freeness criterion in terms of chambers in the category of tame arrangements.
We study the structure of length four polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. The results from this study are used to prove that if $\text{SL}_m(R[X_1,X_2,..., X_n]) = \text{E}_m(R[X_1,X_2,..., X_n])$ for all $n, m \ge 0$ then all length four polynomial automorphisms of $R[X,Y]$ that are conjugates are stably tame.
To solve Nagata's conjecture, Shestakov-Umirbaev constructed a theory for deciding wildness of polynomial automorphisms in three variables. Recently, Karaś and others study multidegrees of polynomial automorphisms as an application of this theory. They give various necessary conditions for triples of positive integers to be multidegrees of tame automorphisms in three variables. In this paper, we prove a strong theorem unifying these results using the generalized Shestakov-Umirbaev theory.
Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal, i.e. $I=(R)$, we compute an upper bound for $°_2(R)$ for some degree function $°_2$ defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of $K^3$ and deduce two elementary proofs of the Jung-van der Kulk Theorem about the tameness of automorphisms of $K^{2}$.
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