On the Tame automorphisms of differential polynomial algebras

1.   Introduction

P. Cohn ^[3] proved that the automorphisms of a free Lie algebra with a finite set of generators are tame. The tameness of the automorphisms of polynomial algebras and free associative algebras in two variables is well known ^[2,6,9,10]. It was proved ^[13,15] that polynomial algebras and free associative algebras in three variables in the case of characteristic zero have wild automorphisms. For example the Nagata automorphism ^[11,13] is a wild automorphism of a free Poisson algebra in three variables. Let $\Delta = \{\delta_1, \ldots, \delta_m\}$ be a basic set of derivation operators and $R\{x_1, x_2, \ldots, x_n\}$ be the polynomial algebra in the variables $x_1, x_2, \ldots, x_n$ over a differential ring R. The basic concepts of differential algebras can be found in ^{[1,5,7,8,12,14]}. The tameness of automorphisms of differential polynomial algebras is studied by B. A. Duisengaliyeva, A. S. Naurazbekova and U. U. Umirbaev ^[4]. They have proved that the tame automorphism group of a differential polynomial algebra over a field of characteristic 0 in two variables with m commuting derivations is a free product with amalgamation. In this paper we give some important subetaoups of the group of differential automorphisms of $R\{x, y\}$ with one derivation operator. Furthermore using the method in Essen ^[16], we prove that the Tame subetaoup of automorphism of $R\{x, y\}$ is the amalgamated free product of the Triangular and the Affine subetaoups over their intersection.

2.   Differential polynomial algebras

Let $R$ be any commutative ring with unity. A mapping $d: R\rightarrow R$ is called a derivation if

holds for all $s, t\in R$.

Let $\Delta = \{\delta_1, \ldots, \delta_m\}$ be a basic set of derivation operators.

A ring $R$ is said to be a differential ring or $\Delta$-ring if all elements of $\Delta$ act on $R$ as a commuting set of derivations, i.e., the derivations $\delta_i: R\rightarrow R$ are defined for all $i$ and $\delta_i\delta_j = \delta_j\delta_i$ for all $i, j$. If $\Delta = \{\delta\}$ (that is, if $\Delta$ consists of only one derivation), then $R$ is called an ordinary differential ring and will be denoted as $\delta$-ring, unless $R$ is called a partial differential ring. In this work, we study on ordinary differential rings.

Let $\Theta$ be the free commutative monoid generated by a derivation operator $\delta$. For $n = 0, 1, 2, ...,$ the elements

of the monoid $\Theta$ are called derivative operators. The order of $\theta$ is defined as $|\theta| = n$. If $R$ is a $\delta$-ring and $R$ is a field (resp. integral domain), then $R$ is called a differential field (resp. integral domain). Let $x$ be a differential indeterminate and let $\Theta x = \{\theta x | \theta\in \Theta\}$ be the set of symbols enumerated by the elements of $\Theta$. Consider the polynomial algebra $R[\Theta x]$ over a $\delta$-ring $R$ generated by the set of (algebraically) independent indeterminates $\Theta x$. It is easy to check that the derivation $\delta$ can be uniquely extended to a derivation of $R[\Theta x]$ by $\delta(\theta x) = ({\delta\theta})x$. Denote this differential ring by $R\{x\}$; it is called the ring of differential polynomials in $x$ over $R$.

By adjoining more differential indeterminates, we obtain the differential ring $R\{x_1, x_2,$ $\ldots, x_n\}$ of the differential polynomials in $x_1, x_2,$ $\ldots, x_n$ over $R$. The ring $R\{x_1, x_2,$ $\ldots, x_n\}$ coincides with the polynomial algebra

Consequently, the set of differential monomials

where $1\leq k_i \leq n$, $\theta_i \in\Theta$, $1\leq i \leq s$, form a linear basis of $R\{x_1, x_2, \ldots, x_n\}$ over $R$.

The degree deg($M$) for a monomial $M = \prod_{i = 1}^s (\theta_i x_{k_i})$, $1\leq k_i \leq n$, $\theta_i \in\Theta$, $1\leq i \leq s$ is defined as in the algebraic case: deg($M$) = s. It is clear that ${\rm{deg}} ({\theta} x_i) = 1$, for each $i$, $\theta \in\Theta$. The elements of the ring $R\{x_1, x_2,$ $\ldots, x_n\}$ are called differential polynomials. The degree of a differential polynomial $f$ is ${\rm{deg}} (f)$ = max$_{M\in f}$${\rm{deg}} (M)$, where $M\in f$ means that $M$ is a differential monomials occurring in $f$. If each term of a differential polynomial $f$ has the same degree, then $f$ is a homogeneous differential polynomial. By ${\rm{deg}} _{x_i}f$ we denote the degree of $f$ with respect to $x_i$ and its derivatives. We have ${\rm{deg}} _{x_i}(\theta x_j) = \delta_{ij}$ where $1\leq i\leq n$, $1\leq j\leq n$, $\theta \in\Theta$ and $\delta_{ij}$ is the Kronecker delta function.

On the other hand one defines the weight of the monomial $M$ as ${\rm{wt}} (M) = \sum_{i = 1}^s |\theta_i|$, where $\theta_i\in\Theta$ and the weight of a differential polynomial $f$ as ${\rm{wt}} (f)$ = max$_{M\in f}$ ${\rm{wt}} (M)$. The terms occurring in $f$ that have the same weight are called isobaric component of $f.$

Let $R$ be a $\delta$-ring, an ideal $\mathit{\boldsymbol{a}}\in R$ is a differential ideal ($\delta$-ideal) if $a\in \mathit{\boldsymbol{a}}$, we have $\delta(a)\in \mathit{\boldsymbol{a}}$. Additionally if $\mathit{\boldsymbol{a}}$ is a radical ideal, then it is called as radical $\delta$-ideal.

A subset $V\subseteq R^2$ is Kolchin-closed if it is the set of all common zeros in $R^2$ of a radical differential ideal $\mathit{\boldsymbol{a}}\subseteq R\{x, y\}$.

Let $R$ be a $\delta$-field, we say that $R$ is differentially closed if every consistent system of differential polynomial equations with coefficients in $R$ has a solution in $R$.

3.   Differential automorphisms of $R\{x, y\}$

Definition 1. Let $R$ be a $\delta$-ring with a derivation operator $\delta$. A differential ring homomorphism, or simply a $\delta$-homomorphism, of $R$ is a ring automorphism $\varphi$ of $R$ such that $\varphi(\delta(a)) = \delta(\varphi(a))$, for all $a \in R$. If the $\delta$-homomorphism $\varphi$ is a ring automorphism of $R$, then $\varphi$ is a $\delta$-automorphism.

Definition 2. Let $K$ be a differential ring extension of a $\delta$-ring $R$. A $\delta$-automorphism $\varphi$ of $K$ is called an $R$-$\delta$-automorphism, provided $\varphi(a) = a$, for all $a\in R$. The set of all $R$-$\delta$-automorphism of K is a group under composition, denoted by ${Aut}_\delta(K|R).$

From now, let $R$ be an integral domain with only derivation operator $\delta$ and $R\{x, y\}$ be the differential polynomial ring in $x, y$ over $R$. Notice that since $R\{x, y\}$ is the free object on the set $\{x, y\}$ in the category of differential $R$-$\delta$-algebras and hence has the universal mapping property.

Definition 3. Let $F_1 = F_1(x, y)$ and $F_2 = F_2(x, y)$ be two differential polynomials in $R\{x, y\}$. A tuple $F = (F_1, F_2)$ in $R\{x, y\}^2$ defines uniquely a $R$-$\delta$-homomorphism $\sigma_F: R\{x, y\}\rightarrow R\{x, y\}$ such that $\sigma_F(x) = F_1$, $\sigma_F(y) = F_2$. For any $P \in R\{x, y\}$, $\sigma_F(P) = P(F_1, F_2)$ (that is, P acts like a bivariate differential operator on pair $(F_1, F_2)).$ Conversely, every $R$-$\delta$-homomorphism $\sigma$ of $R\{x, y\}$(in particular, the inverse of an $R$-$\delta$-automorphism $\sigma \in$ ${Aut} _\delta(R\{x, y\}|R))$ is of form $\sigma_F$ for some tuple $F = (F_1, F_2) \in R\{x, y\}^2.$

Definition 4. A differential polynomial map is a polynomial map $\varphi_F: R^2\rightarrow R^2$ defined by a tuple $F = (F_1, F_2)$ in $R\{x, y\}^2$ such that $\varphi_F(a, b) = (F_1(a, b)$ $, F_2(a, b))$ for any $(a, b) \in R^2$.

Definition 5. A differential polynomial map $\varphi_F$ said to be invertible if there exists $G = (G_1(x, y), G_2(x, y))\in R\{x, y\}^2$ such that $\varphi_F \circ \varphi_G = \mathit{ı}$, where $\mathit{ı}$ is the identity map on the set $R^2$.

The following theorem is a simplified version of a theorem of Kolchin [8, Theorem 4, p. 105] and is analogous to the algebraic fact that in characteristic zero, any two transcendence bases of a field extension $K$ over $k$ have the same cardinality. Kolchin proved the analog of for arbitrary characteristic in terms of differential inseparability basis of a partial differential field extension $K$ over a partial differential field $k$.

Theorem 1. Let $K$ be a differential field extension of an ordinary differential field $k$ of characteristic zero. Then every set $\Sigma \subset K$ that (differentially) generates $K$ over $k$ (that is, $K = k\left\langle \Sigma\right\rangle$) contains a differential transcendence basis $K$ of over $k$ and any two differential transcendence bases of $K$ over $k$ have the same cardinality (called the differential dimension of $K$ over $k$).

The next theorem is known as differential analogue of Hilbert's Nullstellensatz, but is a much deeper result as it involves the notion of a differential closed field.

Theorem 2. Let $R$ be differentially closed field of characteristic zero. The correspondence between the set of radical differential ideals $\textbf{a}$ of $R\{x, y\}$ and the set of Kolchin-closed subset $V\subseteq R^2$, given by

is bijective with the inverse given by

Remark 1. It follows that $V((0)) = R^2$ and $\mathit{\boldsymbol{a}}(R^2) = (0).$ In particular, we have this property : "For any $P\in R\{x, y\}$, if $P$ vanishes for every $(a, b) \in R^2$, then $P = 0$".

From now, for simplicity, we will use $P$ instead of $P(x, y)$ for a differential polynomial $P(x, y)$ of $R\{x, y\}$.

In the next theorem, we show the relation between differential polynomial maps and differential automorphisms. Here when we say that "for some $G$" we mean "for some $G = (G_1, G_2) \in R\{x, y\}^2$".

Theorem 3. Let $R$ be a differential integral domain. Let $F = (F_1, F_2) \in R\{x, y\}^2$.

I. The following are equivalent:

(a) $\sigma_F$ is a differential automorphism with an inverse $\sigma_G$ for some $G$.

(b) $\sigma_F$ is surjective (equivalently, $R\{x, y\} = R\{F_1, F_2\}).$

(c) $x = G_1(F_1, F_2)$ and $y = G_2(F_1, F_2)$ for some $G$.

II. If $F$ satisfies any (and all) of the conditions in I., then

(d) $\varphi_F$ is a invertible differential polynomial map (that is, $\varphi_F\circ \varphi_G = \mathit{ı}$ for some $G$).

(e) $x = F_1(G_1, G_2)$ and $y = F_2(G_1, G_2)$ for some $G.$

(f) $\varphi_F$ has an inverse $\varphi_G$ for some $G$.

III. If $R$ is a differentially closed field of characteristic zero, then (a)–(f) are all equivalent, and the tuple $G\in R\{x, y\}^2$ involved are unique and the same.

Proof. Ⅰ. The implications $(a)\Rightarrow(b)\Rightarrow(c)$ are trivial. We now show the converses: $(c)\Rightarrow(b)\Rightarrow(a).$ From (c), we have $x, y\in R\{F_1, F_2\}$, hence $\sigma_F(R\{x, y\}) = R\{F_1, F_2\} = R\{x, y\},$ which is (b). From (b), let $K$ be the quotient field of $R$ ($R$ is a domain). Then $K$ is a differential field and the two quotient fields $K\langle x, y\rangle$ of $R\{x, y\}$ and $K\langle F_1, F_2\rangle$ of $R\{F_1, F_2\}$ coincide. By Theorem 1, the set $\{F_1, F_2\}$ forms a differential transcendence basis of $K\langle x, y \rangle = K\langle F_1, F_2 \rangle$ over $K.$ So $F_1, F_2$ are differentially algebraic independent over $K$ and a fortiori over $R,$ which means $\sigma_F$ is injective and hence (a).

Ⅱ. We now assume $F$ satisfies (a), (b), and (c) and prove the second part by proving several implications. In this part of the proof, let $G$ be as in (c).

(c)$\Rightarrow$(d):

By (c), for all $(a, b)\in R^2$,

Hence $\varphi_G\circ \varphi_F = \mathit{ı},$ the identity map on $R^2$ and (d) holds.

$(c)\Rightarrow(e)$:

From (c), we have $\sigma_F(F_1(G_1, G_2)) = F_1(G_1(F_1, F_2), G_2(F_1, F_2)) = F_1(x, y) = \sigma_F(x).$ Since $(c)\Rightarrow(a),$ $\sigma_F$ is injective and so $x = F_1(G_1, G_2).$ Similarly, $y = F_2(G_1, G_2)$ and (e) holds.

$(c)\Rightarrow(f)$:

By (c), $\sigma_F \circ \sigma_G = I$, where $I$ is the identity automorphism of $R\{x, y\}$ and from equation (3.1) in $(c)\Rightarrow(d),$ $\varphi_G\circ \varphi_F = \mathit{ı}.$ Since $(c)\Rightarrow(e),$ $\sigma_G\circ \sigma_F = I$ and $\varphi_F\circ \varphi_G = \mathit{ı}.$ Hence (f). In fact, we also prove $\sigma_G$ is the inverse of $\sigma_F.$

Ⅲ. Let $R$ be a differential closed field of characteristic zero. In Ⅰ. we have proved that the $G$ in (c) works for (a) and vice versa. In Ⅱ. we have proved that the $G$ from (c) works for (d), (e), and (f). We now prove the converses and that the $G$ from any of (d), (e), or (f) works for (c), too and hence $G$ is unique (because $G$ defines the inverse of $\sigma_F$) and all $G$ in all the equivalent conditions are the same.

$(d)\Rightarrow(c)$:

Suppose there exists $G = (G_1, G_2)\in R\{x, y\}^2$ such that $\varphi_G\circ \varphi_F = \mathit{ı}$. Then from equation (3.1), the differential polynomials $P = G_1(F_1, F_2)-x$ and $Q = G_2(F_1, F_2)-y$ each vanishes at all $(a, b)\in R^2.$ By Remark 1, $P = Q = 0$, which proves (c).

$(e)\Rightarrow(c)$:

Let $G = (G_1, G_2)$ be as given in (e). We now apply the already proven first two parts of this theorem to $G$ (taking the place of $F$). If we temporarily ornate the corresponding item labels in the theorem for $G$ with $\dagger$, then $(c)^\dagger$ holds using the $F = (F_1, F_2)$ from (e). Since we have proved $(c)^\dagger \Rightarrow (e)^\dagger,$ using the $F$ of $(c)^\dagger,$ it follows that $(e)\Rightarrow(c)$, using the $G$ from (e).

$(f)\Rightarrow(c)$:

Let $G$ be such that $\varphi_G$ is the inverse of $\varphi_F.$ Then of course $\varphi_G\circ \varphi_F = \mathit{ı},$ which is (d), and which implies(c).

4.   The subgroups of the group of differential automorphisms of $R\{x, y\}$

From now let $R$ be a differentially closed field of characteristic zero. Now, for each subset $S\subseteq R\{x, y\}^2$ of interest, we want to find any necessary and sufficient conditions $C_S$ on $F$ that will make $\sigma_F$ an $R$-$\delta$-automorphism of $R\{x, y\}$ for every $F\in S$, that is, for $\sigma_F$ to have an inverse $\sigma_F^{-1} \in Aut_{\delta}(R\{x, y\}/R)$. Thus $C_S$ is not only the set of necessary and sufficient conditions for $\sigma_F$ to be both injective and surjective, but also becomes part of defining properties of the set $S$. Then, it is valid to identify $S$ as a subset $\Sigma(S)$ of $Aut_{\delta}(R\{x, y\}/R)$ via $F\rightarrow \sigma_F$ and not merely as a set of $R$-$\delta$-automorphisms. Moreover, we want to identify those $S\subseteq R\{x, y\}^2$ for which $\Sigma(S) = \{\sigma_F|F\in S\}$ are subetaoups of $Aut_{\delta}(R\{x, y\}/R)$.

Notice that there are two properties $\Sigma(S)$ (or $S$) must satisfy to ensure that $\Sigma(S)$ is a subetaoup. The first of these properties is after showing that $\sigma_F^{-1}$ exists and hence $\sigma_F$, $\sigma_F^{-1}$ $\in Aut_\delta(R\{x, y\}/R)$, it needs to belong to $\Sigma(S)$, that is, $\sigma_F^{-1} = \sigma_G$ for some $G = (G_1, G_2)\in S.$ The second is that for any $\sigma_F$ and $\sigma_G$ in $\Sigma(S)$, their composition $\sigma_F\circ \sigma_G$ must also belong to $\Sigma(S)$. That means $\sigma_F\circ \sigma_G = \sigma_H$ for some $H = (H_1, H_2) \in S$.

Some important subetaoups of $Aut_{\delta}(R\{x, y\}/R)$ are defined as follows:

(1). The Affine differential subetaoup: We determine it as $Aff_\delta(R\{x, y\}/R)$.

Consider $F_i \in R\{x, y\}$, $deg(F_i) = 1,$ $wt(F_i) = 0,$ $(i = 1, 2)$, then for $a, b, c, d, e, f \in R$ put

$F_1 = ax+by+c$, $F_2 = dx+ey+f$ and $det (\begin{bmatrix} a & b \\ d & e \end{bmatrix}) = |JF|$.

After calculating the conditions $C_S$ for $S = Aff_\delta(R\{x, y\}/R)$ and $F = (F_1, F_2)\in S$, we define Affine-$\delta$ subetaoup as follows:

$Aff_\delta(R\{x, y\}/R) = \{(ax+by+c, dx+ey+f)|a, b, c, d, e, f\in R, |JF|\in R^\ast \}$.

(2). The Triangular differential subetaoup: We determine this subetaoup as $J_\delta(R\{x, y\}/R)$ and similarly we calculate it as follows:

$J_\delta(R\{x, y\}/R) = \{(ax+f(y), by+c)|a, b\in R^\ast, c\in R$ and $f(y) \in R\{y\}\}$.

Here $R^\ast$ denotes nonzero elements of $R$.

(3). The Elementary differential subetaoup: We determine this subetaoup as $E_\delta(R\{x, y\}/R)$ and we calculate it as:

$E_\delta(R\{x, y\}/R) = \{(ax+f(y), y)|a\in R^\ast$ and $f(y) \in R\{y\}\}$.

Definition 6. The subetaoup of ${Aut}_\delta(R\{x, y\}|R)$ generated by the Affine $R$-$\delta$-automorphisms and the Triangular $R$-$\delta$-automorphisms is called Tame $\delta$-subetaoup and is denoted by $T_\delta(R\{x, y\}|R).$

Now we will show that $T_\delta(R\{x, y\}|R)$ is the amalgamated free product of Aff$_\delta(R\{x, y\}|R)$ and $J_\delta(R\{x, y\}|R)$ over their intersection.

To prove the results announced above we need the following lemmas.

Lemma 1. Let $G$ be a group generated by two subetaoups $H$ and $K$. Then every element $g$ of $G$ can be written as

for some $\ell\ge1$, where $h_i\in H\setminus\ K$ for all $1\leq i\leq \ell-1$ and $k_i\in K\setminus\ H$ for all $1\leq i\leq \ell$ and $h_0 \in H$.

Proof. See, ^[16], Lemma 5.1.1, p.86.

Let $F = (F_1, F_2), G = (G_1, G_2)\in R\{x, y\}^2$ and let $\sigma_F$ and $\sigma_F$ be automorphisms defined by $F$ and $G$ respectively. We define composition of $\sigma_F$ and $\sigma_G$ as follows: there exists $H = (H_1, H_2)\in R\{x, y\}^2$ such that

To formulate Lemma 2 and Corollary 1 below, we need following notations: let $F = (F_1, F_2)\in R\{x, y\}^2$. Then

Now let $F\in T_\delta(R\{x, y\}|R)$. Then applying Lemma 1 to $G = T_\delta(R\{x, y\}|R)$, $H =$Aff$_\delta (R\{x, y\}|R)$, $K = J_\delta (R\{x, y\}|R)$ and $g = F$ we can write

with $\lambda_i \in \text{Aff} _\delta (R\{x, y\}|R)\setminus J_\delta (R\{x, y\}|R)$ for all $1\leq i\leq \ell-1$ and $\tau_i \in J_\delta (R\{x, y\}|R)\setminus \text{Aff} _\delta (R\{x, y\}|R)$ for all $1\leq i\leq \ell$. Write

Note that for an integer $j$, if $\lambda_j \in \text{Aff} _\delta (R\{x, y\}|R)\setminus J_\delta (R\{x, y\}|R)$, then $d_j = 0$.

Remark 2. Observe that if $\tau \in J_\delta (R\{x, y\}|R)\setminus \text{Aff} _\delta (R\{x, y\}|R)$ then either ${\rm{wt}} (\tau)\geq1$ or ${\rm{wt}} (\tau) = 0$ and ${\rm{deg}} (\tau)\geq2.$

Lemma 2. For any positive integer $\ell$ we have

where for all $1\leq i \leq \ell$, ${\rm{wt}} (\tau_i) = 0$ and the second product is by definition $1$ if $\ell = 1$.

Proof. We prove the lemma by induction on $\ell$. The case $\ell = 1$ is obvious. So let us assume that the statement is true for some $n$ and consider $\text{bideg}(\lambda_{1} \circ \tau_{1} \circ \cdots \circ \lambda_n \circ \tau_n\circ\lambda_{n+1})$. Since $\lambda_{n+1}\notin J_\delta (R\{x, y\}|R)$, we have that $d_{n+1}\neq 0$. Therefore

where $p_i\leq\prod_{j = 1}^{n} \text{deg}(\tau_j(x))$. Finally since $\tau_{n+1} \notin \text{Aff}_\delta(R\{x, y\}|R)$, we have that

which completes the proof.

Corollary 1. $T_\delta(R\{x, y\}|R)$ is the amalgamated free product of $Aff_\delta (R\{x, y\}|R)$ and $J_\delta(R\{x, y\}|R)$ over their intersection, i.e., $T_\delta(R\{x, y\}|R)$ is generated by these two groups and if $\tau_i\in J_\delta(R\{x, y\}|R)\setminus Aff_\delta(R\{x, y\}|R)$ and $\lambda_i\in Aff_\delta(R\{x, y\}|R)\setminus J_\delta(R\{x, y\}|R)$ then $\tau_1 \circ\lambda_2 \cdots\circ \tau_{n-1} \circ\lambda_n\circ\tau_n$ does not belong to $Aff_\delta(R\{x, y\}|R)$.

Proof. By the definition, $T_\delta(R\{x, y\}|R)$ is generated by Aff$_\delta(R\{x, y\}|R)$ and $J_\delta(R\{x, y\}|R)$. So suppose that

with $\tau_i\in J_\delta(R\{x, y\}|R)\setminus$Aff$_\delta(R\{x, y\}|R)$ and $\lambda_i\in Aff_\delta(R\{x, y\}|R)\setminus J_\delta(R\{x, y\}|R)$ for all i. Then

Therefore we have

Since $\tau_i\in J_\delta(R\{x, y\}|R)\setminus$ Aff$_\delta(R\{x, y\}|R)$, then

Now let us suppose for all $i$, ${\rm{wt}} (\tau_i) = 0$ so ${\rm{deg}} (\tau_i)\geq2$. By the Lemma 2, we have

From the equation (4.1), ${\rm{deg}} (\tau_n) = 1$ which is a contradiction. Let us suppose that for all $i$, ${\rm{wt}} (\tau_i)\geq1$, so

which is a contradiction by (4.2).

Acknowledgments

The Theorem 3 is by, and included with permission from William Sit through private communications. The authors would like to thank Professor William Sit and Professor Ualbai Umirbaev for their supports and valuable comments and suggestions to improve the quality of the paper.

Conflict of interest

The authors declare no conflict of interest.