Generalization of a counterexample to Derksen’s theorem in characteristic two

Abstract

The Derksen group is the subgroup of the tame subgroup which is generated by affine automorphisms and one particular non-linear automorphism. It is known that the Derksen group is equal to the entire tame subgroup in characteristic zero and in dimension greater than or equal to three (Derksen’s Theorem). Recently, Maubach and Willems (Serdica Math J 37:305–322, 2011) proved that if the defining field of polynomial automorphisms is prime field of characteristic two, then Derksen’s Theorem does not hold in dimension three. In this paper, we generalize the above counterexample to the case of higher dimension.

1 Introduction

Throughout the paper, we use ${\mathbb {K}}$ to denote a field of characteristic $p := \text {char}({\mathbb {K}}) \ge 0$ and ${\mathbb {K}}[{X_1}, \ldots , {X_n}]$ denotes the polynomial ring in n indeterminates ${X_1}, \ldots , {X_n}$ over ${\mathbb {K}}$. We denote by $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ the general automorphism group of ${{\,\mathrm{Spec}\,}}{\mathbb {K}}[{X_1}, \ldots , {X_n}]$ over ${{\,\mathrm{Spec}\,}}{\mathbb {K}}$. $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ is anti-isomorphic to $\mathop {\mathrm {Aut}_{{\mathbb {K}}}} {\mathbb {K}}[{X_1}, \ldots , {X_n}]$ (the group of ${\mathbb {K}}$-automorphisms of ${\mathbb {K}}[{X_1}, \ldots , {X_n}]$). We identify $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ with $\mathop {\mathrm {Aut}_{{\mathbb {K}}}} {\mathbb {K}}[{X_1}, \ldots , {X_n}]$ via the above correspondence (See van den Essen 2000, Introduction for details). The subgroup of $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ of affine automorphisms is denoted by $\mathop {\mathrm {Aff}}_{n}({\mathbb {K}})$, and the subgroup of $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ generated by elementary automorphisms is denoted by $\mathop {\mathrm {EA}}_{n}({\mathbb {K}})$. We recall that ${\mathop {\mathrm {Aff}}}_{n}({\mathbb {K}}) \cong {\mathbb {K}}^{n} \rtimes \mathop {\mathrm {GL}_{n}}({\mathbb {K}})$. Let $\mathop {\mathrm {TA}_{n}}({\mathbb {K}})$ be the sugroup of $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$ generated by two subgroups $\mathop {\mathrm {Aff}}_{n}({\mathbb {K}})$ and $\mathop {\mathrm {EA}}_{n}({\mathbb {K}})$. $\mathop {\mathrm {TA}_{n}}({\mathbb {K}})$ is called the tame subgroup.

Set

$$\begin{aligned} \epsilon := ({X_1} + {X_2^2}, {X_2}, {X_3}, \ldots , {X_n}) \in \mathop {\mathrm {EA}_{n}}({\mathbb {K}}), \end{aligned}$$

and

$$\begin{aligned} \mathop {\mathrm {DA}_{n}}({\mathbb {K}}) := \langle {\mathop {\mathrm {Aff}}}_{n}({\mathbb {K}}), \epsilon \rangle \subset \mathop {\mathrm {TA}_{n}}({\mathbb {K}}). \end{aligned}$$

$\mathop {\mathrm {DA}_{n}}({\mathbb {K}})$ is called the Derksen group. It is known that if ${\mathbb {K}}$ is a commutative ${\mathbb {Q}}$-algebra, then the equation

$$\begin{aligned} \mathop {\mathrm {DA}_{n}}({\mathbb {K}}) = \mathop {\mathrm {TA}_{n}}({\mathbb {K}}) \end{aligned}$$

holds for each $n \ge 3$ (van den Essen 2000, Theorem 5.2.1). The result is due to Harm Derksen and is called Derksen’s Theorem (See van den Essen 2000, Section 5.2 for details). Bodnarchuk generalized Derksen’s Theorem in $\text {char}(\mathbb {K}) = 0$ (Bodnarchuk 2005). On the other hand, Maubach and Willems proved that Derksen’s Theorem in positive characteristic is not true for a special case (Maubach and Willems 2011). More precisely, they showed that if ${\mathbb {K}} = {\mathbb {F}}_{2}$, then $\mathop {\mathrm {DA}_{3}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{3}}({\mathbb {F}}_{2})$ holds (Maubach and Willems 2011, Lemma 2.2). Here, we denote by ${\mathbb {F}}_{q}$ a finite field with q elements ($p = \text {char}({\mathbb {F}}_{q})$, $q = p^r$, and $r \ge 1$). In this paper, we generalize the above counterexample (Maubach and Willems 2011, Lemma 2.2) to the case of higher dimension, namely, if ${\mathbb {K}} = {\mathbb {F}}_{2}$, then Derksen’s Theorem does not hold for each $n \ge 3$.

2 Main results

Throughout the section, the symmetric group on a finite set ${\mathcal {A}}$ and the alternating group on ${\mathcal {A}}$ are denoted by $\mathop {\mathrm {Sym}}({\mathcal {A}})$ and $\mathop {\mathrm {Alt}}({\mathcal {A}})$, respectively. We remark that there exists a natural map

$$\begin{aligned} \begin{array}{ccccc} {\pi }_{q} :&\mathop {\mathrm {GA}_{n}}({\mathbb {F}}_{q})\rightarrow & {} \mathop {\mathrm {Sym}}({{\mathbb {F}}_{q}^n}), \end{array} \end{aligned}$$

(see Maubach and Willems 2011, Definition 1.4 for details).

In this section, we prove the main theorem (Theorem 1) of this paper. Before we prove Theorem 1, we recall a result from (Hakuta 2018, Corollary 2) (this is a corollary of Hakuta 2018, Main Theorem 2) which gives a sufficient condition for the containment ${\pi }_{q}({\mathop {\mathrm {Aff}}}_{n}({\mathbb {F}}_{q})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{q}^n})$.

Corollary 1

(Hakuta 2018, Corollary 2) If $q = 2^{m}$, $m \ge 2$, and $n \ge 2$, or $q = 2$ and $n \ge 3$ then we have ${\pi }_{q}({\mathop {\mathrm {Aff}}}_{n}({\mathbb {F}}_{q})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{q}^n})$.

Thanks to (Hakuta 2018, Corollary 2) and (Maubach 2001, Theorem 2.3(ii)), the same arguments as in (Maubach and Willems 2011, Lemma 2.2) can be used to prove the main theorem (Theorem 1).

Theorem 1

(Main Theorem) If $n \ge 3$ then $\mathop {\mathrm {DA}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})$. In other words, Derksen’s Theorem is not true in the case where ${\mathbb {K}} = {\mathbb {F}}_{2}$ and $n \ge 3$.

Proof

Since ${\pi }_{2}({\epsilon }) = {\pi }_{2}(({X_1} + {X_2}, {X_2}, \ldots , {X_n})) \in {\pi }_{2}(\mathop {\mathrm {Aff}_{n}}({\mathbb {F}}_{2}))$, we have ${\pi }_{2}(\mathop {\mathrm {Aff}_{n}}({\mathbb {F}}_{2})) = {\pi }_{2}(\langle \mathop {\mathrm {Aff}_{n}}({\mathbb {F}}_{2}), \epsilon \rangle ) = {\pi }_{2}(\mathop {\mathrm {DA}_{n}}({\mathbb {F}}_{2}))$. Then by (Hakuta 2018, Corollary 2), the containment ${\pi }_{2}({\mathop {\mathrm {Aff}}}_{n}({\mathbb {F}}_{2})) = {\pi }_{2}(\mathop {\mathrm {DA}_{n}}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n})$ holds for each $n \ge 3$. On the other hand, from (Maubach 2001, Theorem 2.3(ii)), we have ${\pi }_{2}(\mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})) = \mathop {\mathrm {Sym}}({{\mathbb {F}}_{2}^n})$. Hence we obtain

$$\begin{aligned} {\pi }_{2}({\mathop {\mathrm {DA}}}_{n}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n}) \subsetneq \mathop {\mathrm {Sym}}({{\mathbb {F}}_{2}^n}) = {\pi }_{2}(\mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})). \end{aligned}$$

Thus we must have $\mathop {\mathrm {DA}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})$. This completes the proof. $\square$

The rest of this section is devoted to the description of Meister’s Linearization problem in ${\mathbb {F}}_{2}$. We refer the reader to van den Essen (2000), Maubach and Poloni (2009) and Maubach and Willems (2011) for details of Meister’s Linearization problem.

In (Maubach and Willems 2011, Section 4) Maubach and Willems compared two subgroups of $\mathop {\mathrm {GA}_{n}}({\mathbb {K}})$. These subgroups are $\mathop {\mathrm {GLIN}_{n}}({\mathbb {K}})$ and $\mathop {\mathrm {GTAM}_{n}}({\mathbb {K}})$ (See below for the definition). Here, we reconsider these subgroups.

Definition 1

(Maubach and Willems 2011, Definition 4.1) Let G be a group, and H a subgroup of G. We define ${\mathcal {N}}(H, G)$ to be the smallest normal subgroup of G that contains H, i.e., ${\mathcal {N}}(H, G) = \langle g^{-1}hg \; \vert \; h \in H, g \in G \rangle$.

Set

$$\begin{aligned}&\mathop {\mathrm {GLIN}_{n}}({\mathbb {K}}) := {\mathcal {N}}(\mathop {\mathrm {GL}_{n}}({\mathbb {K}}), \mathop {\mathrm {GA}_{n}}({\mathbb {K}})), \end{aligned}$$

(2.1)

$$\begin{aligned}&\mathop {\mathrm {GTAM}_{n}}({\mathbb {K}}) := {\mathcal {N}}(\mathop {\mathrm {TA}_{n}}({\mathbb {K}}), \mathop {\mathrm {GA}_{n}}({\mathbb {K}})), \end{aligned}$$

(2.2)

and

$$\begin{aligned} \mathop {\mathrm {TLIN}_{n}}({\mathbb {K}}) := {\mathcal {N}}(\mathop {\mathrm {GL}_{n}}({\mathbb {K}}), \mathop {\mathrm {TA}_{n}}({\mathbb {K}})). \end{aligned}$$

(2.3)

Maubach and Willems obtained the following result about two subgroups $\mathop {\mathrm {GLIN}_{n}}({\mathbb {K}})$ and $\mathop {\mathrm {GTAM}_{n}}({\mathbb {K}})$.

Theorem 2

(Maubach and Willems 2011, Theorem 4.2) If $n \ge 2$ and ${\mathbb {K}} = {\mathbb {F}}_{2}$ then $\mathop {\mathrm {GLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})$, and hence $\mathop {\mathrm {TLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})$.

Here, we give an alternative proof of Theorem 2 (Maubach and Willems 2011, Theorem 4.2) in the case where $n \ge 3$ by using the same argument as in the proof of Theorem 1.

Alternative Proof of Theorem 2

(Maubach and Willems 2011, Theorem 4.2).

From the definitions of the above three subgroups (2.1), (2.2), and (2.3), it is easy to see that $\mathop {\mathrm {GLIN}_{n}}({\mathbb {F}}_{2}) \subset \mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})$ and $\mathop {\mathrm {TLIN}_{n}}({\mathbb {F}}_{2}) \subset \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})$. Suppose that $n \ge 3$. We prove

$$\begin{aligned} \mathop {\mathrm {GLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2}) \quad \text {and} \quad \mathop {\mathrm {TLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2}). \end{aligned}$$

By (Hakuta 2018, Corollary 2) , we have

$$\begin{aligned} {\pi }_{2}(\mathop {\mathrm {GLIN}_{n}}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n}) \quad \text {and} \quad {\pi }_{2}(\mathop {\mathrm {TLIN}_{n}}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n}). \end{aligned}$$

It is obviously true that $\mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2}) \subset \mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})$. From (Maubach 2001, Theorem 2.3(ii)), we have ${\pi }_{2}(\mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})) = {\pi }_{2}(\mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})) = \mathop {\mathrm {Sym}}({{\mathbb {F}}_{2}^n})$. Hence we obtain

$$\begin{aligned} {\pi }_{2}({\mathop {\mathrm {GLIN}}}_{n}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n}) \subsetneq \mathop {\mathrm {Sym}}({{\mathbb {F}}_{2}^n}) = {\pi }_{2}(\mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})), \end{aligned}$$

and

$$\begin{aligned} {\pi }_{2}({\mathop {\mathrm {TLIN}}}_{n}({\mathbb {F}}_{2})) \subset \mathop {\mathrm {Alt}}({{\mathbb {F}}_{2}^n}) \subsetneq \mathop {\mathrm {Sym}}({{\mathbb {F}}_{2}^n}) = {\pi }_{2}(\mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})). \end{aligned}$$

Thus $\mathop {\mathrm {GLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {GTAM}_{n}}({\mathbb {F}}_{2})$ and $\mathop {\mathrm {TLIN}_{n}}({\mathbb {F}}_{2}) \subsetneq \mathop {\mathrm {TA}_{n}}({\mathbb {F}}_{2})$ hold for each $n \ge 3$. $\square$