Higher derivations of Jacobian type in positive characteristic

TL;DR

This paper investigates higher derivations of Jacobian type in positive characteristic, providing conditions for polynomial extendability and characterizations of variables and univariate polynomials.

Contribution

It introduces necessary and sufficient conditions for extendability of polynomial tuples and characterizes variables using higher derivations in positive characteristic.

Findings

Conditions for extendability of polynomial tuples

Characterizations of variables and univariate polynomials

Insights into higher derivations in positive characteristic

Abstract

In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral domain $R$ of positive characteristic. In particular, we give characterizations of variables and univariate polynomials by using the terms of higher derivations of Jacobian type in the polynomial ring in two variables over a field of positive characteristic.

Tables

$\ell$	$\ell !=p^{e\left(\ell \right)}{m}_{\ell }$	${\left[d_1\right]}^{\ell }\left(x\right)$	$\left({\ell !}^{-1}{\left[d_1\right]}^{\ell }\right)\left(x\right)$
$1$	$3^0\cdot 1$	$3y^2$	$0$
$2$	$3^0\cdot 2$	$3\cdot 2y$	$0$
$3$	$3^1\cdot 2!$	$3!$	$1$
$\ell \ge 4$	$\ell !$	$0$	$0$

Equations

$${\phi }_D(f)=\sum _{i=0}^{\infty }{D}_i(f)t^i$$

$$B^D=\bigcap _{\ell \ge 1}\ker D_{\ell }.$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\iota }_0:k{\cong }_k\kappa (0)\to k[x,y]{\otimes }_{k[xy]}\kappa (0){\cong }_{k}k[x,y]/(xy).$$

$$k[x,y]/(xy)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0=v(xy)=v(x)v(y)=t^2,$$

$$[d{]}^{\ell }(b)=[d]\left([d{]}^{\ell -1}(b)\right).$$

$${\Delta }_F(g)=\det {\left(\partial x_j(f_i)\right)}_{1\le i,j\le n},{\tilde{\Delta }}_F(g)=[\det ]{\left(\left[\partial x_j\right](f_i)\right)}_{1\le i,j\le n},$$

$${\phi }_{\ell }(g)=\sum _{i=0}^{\ell }\frac{1}{i!}[{\tilde{\Delta }}_F{]}^i(g)t^i.$$

$$N_r^2=0,C_r/N_r{\cong }_{B}B[t]/(t^{p+r-1}).$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\phi }_{p+r}(g)=\sum _{\ell =0}^{p-1}\frac{1}{\ell !}{[{\tilde{\Delta }}_F]}^{\ell }(g)t^{\ell }+\sum _{i=0}^{r-1}{D}_{p+i}(g)t^{p+i}$$

$$\phi (g)=\sum _{\ell =0}^{p-1}\frac{1}{\ell !}{[{\tilde{\Delta }}_F]}^{\ell }(g)t^{\ell }+\sum _{i=0}^{\infty }{D}_{p+i}(g)t^{p+i}.$$

$$\phi (x)=\sum _{\ell =0}^{\infty }x{t}^{\ell },\phi (y)=y+yt.$$

$$[d{]}^i(g)=p^{e(i)}{g}_i.$$

$$\mathrm{Exp}(td)(g)=\sum _{\ell =0}^{\infty }\frac{[d{]}^{\ell }}{\ell !}(g)t^{\ell }=\sum _{\ell =0}^{\infty }\frac{1}{m_{\ell }}{g}_{\ell }{t}^{\ell }.$$

$${[d_2]}^{\ell }(x)=\{\begin{array}{ll}(-1{)}^{\frac{\ell +1}{2}}y& (\mathrm{\ell }\mathrm{is}\mathrm{odd}),\\ (-1{)}^{\frac{\ell }{2}}x& (\mathrm{\ell }\mathrm{is}\mathrm{even}),\end{array}$$

$$\mathrm{Exp}(t\cdot {}^{\sigma }d)={\sigma }^{-1}\circ \mathrm{Exp}(td)\circ \sigma .$$

$${\phi }_{D_2}(x)=x,{\phi }_{D_2}(y)=y+t,$$

Full text

Research of the author was partially supported by Grant-in-Aid for JSPS Fellows (No. 18J10420) from Japan Society for the Promotion of Science

0. Introduction

In this paper, we study higher derivations of Jacobian type in positive characteristic. In the case where the characteristic of the ground field is zero, derivations of Jacobian type are well known and they are one of the most important tools for understanding polynomial rings. See e.g., [6], [2] and [4]. However, in the case where the characteristic of the ground field is positive, there are no concepts corresponding to derivations of Jacobian type.

In Section 1, we recall some kinds of higher derivations and their properties. Also we recall variables, univariate polynomials and extendable $(n-1)$-tuples of polynomials.

In Section 2, we recall some properties of a smooth extension of rings. In Definition 2.4, we introduce concepts for higher derivations of Jacobian type. We show that smooth ring extensions guarantee the existence of higher derivations of Jacobian type (Proposition 2.5). The main result in this paper is Theorem 2.8 which gives a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable by the terms of higher derivations of Jacobian type. This is a generalization of [1, Proposition 2.3] in positive characteristic.

In Section 3, we study higher derivations of Jacobian type on $k[x,y]$. In Theorem 3.1 and Corollary 3.2, we give characterizations of variables and univariate polynomials by using the terms of higher derivations of Jacobian type. Theorem 3.1 is a generalization of [3, Theorem 3.2] in the case where the characteristic of the ground field is positive.

1. Preliminaries

Let $R$ be an integral domain of characteristic $p\geq 0$. For a positive integer $n\geq 1$, we denote $R^{[n]}$ by the polynomial ring in $n$ variables over $R$ and $Q(R)$ by the field of fractions.

Through this section, assume that $B$ is an integral domain containing $R$. Let $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ be a family of $R$-linear maps $D_{\ell}:B\to B$ for $\ell\geq 0$. We say that $D$ is a higher $R$-derivation on $B$ if, for $f,g\in B$ and $\ell\geq 0$,

1. (a)

$D_{0}={\rm id}_{B}$, 2. (b)

$\displaystyle D_{\ell}(fg)=\sum_{i+j=\ell}D_{i}(f)D_{j}(g)$.

Note that, for a higher $R$-derivation $D=\{D_{\ell}\}_{\ell=0}^{\infty}$, $D_{1}$ is an $R$-derivation on $B$.

For a higher $R$-derivation $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ on $B$, we define the map $\varphi_{D}:B\to B[[t]]$, where $B[[t]]$ is the formal power series ring in one variable over $B$, by

[TABLE]

for $f\in B$. The above condition (b) implies that $\varphi_{D}$ is a homomorphism of $R$-algebras, condition (a) implies that $\varphi_{D}(f)|_{t=0}=f$. We call the mapping $\varphi_{D}$ the homomorphism associated to $D$. We denote $B^{D}$ by the intersections of the kernel of $D_{\ell}$ for $\ell\geq 1$, that is,

[TABLE]

We say that $D$ is trivial if $B^{D}=B$. A higher $R$-derivation $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ on $B$ is locally finite if $D$ satisfies:

1. (c)

for any $f\in B$, there exists a positive integer $N_{f}\geq 1$ such that $D_{\ell}(f)=0$ for any $\ell\geq N_{f}$,

and is iterative if $D$ satisfies:

1. (d)

$\displaystyle D_{i}\circ D_{j}=\binom{i+j}{j}D_{i+j}$ for any $i,j\geq 0$.

When $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ satisfies the above conditions (a), (b), (c) and (d), we say $D$ is a locally finite iterative higher $R$-derivation, for short an lfihd.

Let $D$ be an lfihd on $B$. An element $s$ of $B$ is called a local slice of $D$ if it satisfies the following conditions:

1. (a)

$s\not\in B^{D}$, 2. (b)

$\deg_{t}(\varphi_{D}(s))={\rm min}\{\deg_{t}(\varphi_{D}(b))\ |\ b\in B\setminus B^{D}\}$.

Here, we note that every nontrivial lfihd on $B$ has local slices. A local slice $s\in B$ of $D$ is called a slice if the leading coefficient of $\varphi_{D}(s)$ is a unit of $B$.

Proposition 1.1**.**

(cf. [7, Lemma 1.4])* Let $D$ be an lfihd on $B$. If $D$ has a slice $s\in B$, then $B=B^{D}[s]$ and $s$ is indeterminate over $B^{D}$.*

In the rest of this section, we assume that $B=R[x_{1},\ldots,x_{n}]\cong_{R}R^{[n]}$ is the polynomial ring in $n$ variables over $R$. Let $f\in B\setminus R$ be a non-constant polynomial. $f$ is called a variable (or coordinate) over $R$ if $R[f]^{[n-1]}=B$. Finally, $f$ is called univariate over $R$ if there exists a variable $g\in B$ such that $f\in R[g]$. An $(n-1)$-tuple polynomials $f_{1},\ldots,f_{n-1}\in B$ is said to be extendable if $R[f_{1},\ldots,f_{n-1}]^{[1]}=B$.

2. Higher derivations of Jacobian type in positive characteristic

First of all, we prepare some notation and results of general commutative ring theory. Let $A$ be a commutative ring and let $B$ be a commutative $A$-algebra via a homomorphism $\varphi:A\to B$. We say that $B$ is smooth over $A$ if for any $A$-algebra $C$ with $g:A\to C$, an ideal $N\subset C$ with $N^{2}=0$ and a homomorphism of $A$-algebras $u:B\to C/N$, there exists a homomorphism of $A$-algebras $v:B\to C$ such that $v\circ\pi=u$, where $\pi:C\to C/N$ is the natural homomorphism. That is, $v$ commutes the following diagram:

[TABLE]

For $\mathfrak{p}\in\operatorname{Spec}A$, we denote the residue field by $\kappa(\mathfrak{p})=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$.

Proposition 2.1**.**

Let $\varphi:A\to B$ be a homomorphism of commutative rings. For $\mathfrak{p}\in\operatorname{Spec}A$, let $\iota_{\mathfrak{p}}:\kappa(\mathfrak{p})\to B\otimes_{A}\kappa(\mathfrak{p})$ be the natural homomorphism of $A$-algebras. If $\varphi$ is smooth, then $\iota_{\mathfrak{p}}$ is also smooth for any $\mathfrak{p}\in\operatorname{Spec}A$.

Proof.

Omitted. ∎

Example 2.2**.**

Let $R$ be an integral domain containing a prime field and let $R[x,y]\cong_{R}R^{[2]}$. Then the following assertions hold true.

1. (a)

The natural inclusion $R\to R[x,y]$ is smooth. 2. (b)

For a variable $f\in R[x,y]$, the natural inclusion $R[f]\to R[x,y]$ is smooth. 3. (c)

For $xy\in R[x,y]$, the natural inclusion $R[xy]\to R[x,y]$ is not smooth.

Proof.

(a) and (b) are obvious. We prove the assertion (c).

Assume to the contrary that $\iota:R[xy]\to R[x,y]$ is smooth. Let $k$ be an algebraic closure of $Q(R)$. By Proposition 2.1, the following homomorphism $\iota_{0}$ is smooth:

[TABLE]

Set $C=k[t]/(t^{3})$ and $N=t^{2}C$, where $k[t]\cong_{k}k^{[1]}$. Then $N^{2}=0$. We define $u:k[x,y]/(xy)\to C/N$ by $u(x)=u(y)=t$. Since $\iota_{0}$ is smooth, there exists a homomorphism $v:k[x,y]/(xy)\to C$ of $k$-algebras such that $\pi\circ v=u$, namely, $v$ commutes the following diagram:

[TABLE]

Then $v(x)=t+at^{2}$ and $v(y)=t+bt^{2}$ for some $a,b\in k$. However,

[TABLE]

which is a contradiction. ∎

From now on, let $R$ be an integral domain containing a prime field of characteristic $p\geq 0$ and let $B=R[x_{1},\ldots,x_{n}]\cong_{R}R^{[n]}$ be the polynomial ring in $n$ variables over $R$. We denote $\partial_{x_{i}}$ by the partial derivative with respect to $x_{i}$.

Definition 2.3**.**

Let $d$ be an $R$-linear map on $B$. For $b\in B$, we denote $[d](b)$ by the result of calculation of $d(b)$ as if the characteristic of $R$ is zero, that is, we consider $p\neq 0$ in $B$. For $\ell\geq 1$, we define, for $b\in B$,

[TABLE]

For $f_{1},\ldots,f_{n-1}\in B$, let $F=(f_{1},\ldots,f_{n-1})$ and $R[F]=R[f_{1},\ldots,f_{n-1}]$. Then $F$ defines the $R$-derivation $\Delta_{F}$ and $\widetilde{\Delta}_{F}$ on $B$ by, for $g\in B$,

[TABLE]

where for a matrix $A$, $[{\rm det}]\>(A)$ means the result of calculation of ${\rm det}\>(A)$ as if the characteristic of $R$ is zero. $\widetilde{\Delta}_{F}$ is called the Jacobian derivation determined by $F$.

Definition 2.4**.**

A higher $R$-derivation $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ on $B$ is of Jacobian type if there exists $F=(f_{1},\ldots,f_{n-1})\in B^{n-1}$ such that

1. (a)

$R[F]\cong_{R}R^{[n-1]}$, 2. (b)

$f_{1},\ldots,f_{n-1}\in B^{D}$, 3. (c)

$D_{1}=\widetilde{\Delta}_{F}$, 4. (d)

$\displaystyle D_{\ell}=\frac{1}{\ell!}[D_{1}]^{\ell}$ for  $\begin{cases}{\rm any}\>\>\ell\geq 0&{\rm if}\>\>p=0,\\ 0\leq\ell\leq p-1&{\rm if}\>\>p>0.\end{cases}$

Note that the above condition (b) is equivalent to $\varphi_{D}(f_{i})=f_{i}$ for $1\leq i\leq n-1$, that is, $\varphi_{D}$ is a homomorphism over $R[F]$.

Proposition 2.5**.**

For $f_{1},\ldots,f_{n-1}\in B\setminus R[{x_{1}}^{p},\ldots,{x_{n}}^{p}]$, let $F=(f_{1},\ldots,f_{n-1})$. Suppose that $R[F]\cong_{R}R^{[n-1]}$. If the natural inclusion $R[F]\to B$ is smooth, then there exists a higher $R$-derivation $D$ on $B$ of Jacobian type determined by $F$.

Proof.

Define $D_{0}={\rm id}_{B}$ and $D_{1}=\widetilde{\Delta}_{F}\>(\neq 0)$. Let $B[t]\cong_{B}B^{[1]}$. Here, we define a map $\varphi_{\ell}:B\to B[t]/(t^{\ell+1})$ by, for $g\in B$ and $1\leq\ell\leq p-1$,

[TABLE]

Then $\varphi_{\ell}$ is a homomorphism of $R[F]$-algebras such that $\varphi_{\ell}(g)|_{t=0}=g$ for any $g\in B$.

For $r\geq 0$, let $C_{r}=B[t]/(t^{p+r})$ and $N_{r}=t^{p+r-1}C_{r}$. Then

[TABLE]

Since $R[F]\to B$ is smooth, there exists a homomorphism $\varphi_{p+r}:B\to C_{r}$ of $R[F]$-algebras such that $\pi_{r}\circ\varphi_{p+r}=\varphi_{p+r-1}$, that is, we have the following diagram:

[TABLE]

Moreover $\varphi_{p+r}(g)|_{t=0}=g$ for any $g\in B$. For $0\leq i\leq r-1$, by using $\varphi_{p+r}$, we define a homomorphism of $R$-modules $D_{p+i}:B\to B$ by the following formula:

[TABLE]

for $g\in B$. By constructing such homomorphisms inductively, we have a homomorphism of $R[F]$-algebras $\varphi=\varphi_{\infty}:B\to B[[t]]\cong_{B}B^{[[1]]}$ such that, for $g\in B$, $\varphi(g)|_{t=0}=g$ and

[TABLE]

Set $D_{\ell}={\ell!}^{-1}{[\widetilde{\Delta}_{F}]}^{\ell}$ for $0\leq\ell\leq p-1$ and $D=\{D_{\ell}\}_{\ell=0}^{\infty}$. By the construction of each $D_{\ell}$, we see that $D$ is a higher $R$-derivation on $B$ of Jacobian type determined by $F$. ∎

We note that smoothness is not necessarily for the existence of higher $R$-derivations of Jacobian type. We give an example below.

Example 2.6**.**

Let $f=xy\in\mathbb{F}_{2}[x,y]\cong_{\mathbb{F}_{2}}\mathbb{F}_{2}^{[2]}$. We define $\varphi:\mathbb{F}_{2}[x,y]\to\mathbb{F}_{2}[x,y][[t]]$ by

[TABLE]

Then there exists a higher $R$-derivation $D=\{D_{\ell}\}_{\ell=0}^{\infty}$ on $\mathbb{F}_{2}[x,y]$ such that $\varphi_{D}=\varphi$ and $D_{1}=\widetilde{\Delta}_{f}$. Moreover, $xy\in\mathbb{F}_{2}[x,y]^{D}$. Therefore $D$ is a higher $R$-derivation of Jacobian type determined by $f$, but the natural inclusion $\mathbb{F}_{2}[xy]\to\mathbb{F}_{2}[x,y]$ is not smooth. **

In order to explain the statement of the main theorem (Theorem 2.8), we introduce some definitions as below. For a positive integer $\ell\geq 1$, we write $\ell!=p^{e(\ell)}m_{\ell}$, where $p$ does not divide $m_{\ell}$. Let $\ell\geq 1$ and $g\in B$. For a non-zero $R$-derivation $d\in\operatorname{Der}_{R}B$, we say that ${\ell!}^{-1}[d]^{\ell}$ is defined at $g$ if, for any $1\leq i\leq\ell$, there exists $g_{i}\in B$ such that

[TABLE]

We define its value by $({\ell!}^{-1}[d]^{\ell})(g)=m_{\ell}^{-1}g_{\ell}$. When ${\ell!}^{-1}[d]^{\ell}$ is defined at any $g\in B$ and $\ell\geq 1$, we consider the map ${\rm Exp}(td):B\to B[[t]]\cong_{B}B^{[[1]]}$ defined by

[TABLE]

By the definition of ${\rm Exp}(td)$, we see that it is a homomorphism of $R$-algebras and satisfies that ${\rm Exp}(td)(g)|_{t=0}=g$ for $g\in B$. Therefore, there exists a higher $R$-derivation $D$ on $B$ such that ${\rm Exp}(td)=\varphi_{D}$.

In order to check whether the map ${\rm Exp}(td)$ is defined or not, it is enough to show that ${\ell!}^{-1}[d]^{p^{\ell}}$ is defined at $x_{1},\ldots,x_{n}$ for any $\ell\geq 1$.

Example 2.7**.**

Let $B=\mathbb{F}_{3}[x,y]\cong_{\mathbb{F}_{3}}\mathbb{F}_{3}^{[2]}$. Set $d_{1}=\widetilde{\Delta}_{x-y^{3}}=3y^{2}\partial_{x}+\partial_{y}$ and $d_{2}=\widetilde{\Delta}_{xy}=-y\partial_{x}+x\partial_{y}$. Then ${\rm Exp}(td_{1})$ is defined, but ${\rm Exp}(td_{2})$ is not defined. Indeed, for $d_{1}$, it is clear that ${\ell!}^{-1}{[d_{1}]}^{\ell}$ is defined at $y$ for $\ell\geq 1$. Also, it is defined at $x$ for $\ell\geq 1$ as the following table.

Therefore ${\rm Exp}(td_{1})$ can be defined and ${\rm Exp}(td_{1})(x)=x+t^{3}$, ${\rm Exp}(td_{1})(y)=y+t$. Moreover, it is easy to show that ${\rm Exp}(td_{1})(x-y^{3})=x-y^{3}$.

On the other hand, for $d_{2}$,

[TABLE]

hence ${\ell!}^{-1}{[d_{2}]}^{\ell}$ is not defined at $x$ when $\ell\geq 2$. **

Let $d\in\operatorname{Der}_{R}B$ be a non-zero $R$-derivation. We consider $\operatorname{Aut}_{R}B$ as a subgroup of $\operatorname{Aut}_{R}B[[t]]$ by $\sigma(t)=t$ for $\sigma\in\operatorname{Aut}_{R}B$. If ${\rm Exp}(td)$ can be defined, then ${\rm Exp}(t\cdot{}^{\sigma}d)$ can be defined for any $\sigma\in\operatorname{Aut}_{R}B$, where ${}^{\sigma}d:=\sigma^{-1}\circ d\circ\sigma$. In particular, the following holds:

[TABLE]

The following is the main result in this paper which is a generalization of [1, Proposition 2.3] in positive characteristic.

Theorem 2.8**.**

Let $R$ be an integral domain of characteristic $p>0$ and let $B=R[x_{1},\ldots,x_{n}]\cong_{R}R^{[n]}$ be the polynomial ring in $n$ variables over $R$. For $f_{1},\ldots,f_{n-1}\in B$, let $F=(f_{1},\ldots,f_{n-1})$. Then the following conditions are equivalent:**

1. (i)

$F$* is extendable.* 2. (ii)

$D=\{{\ell!}^{-1}{[\widetilde{\Delta}_{F}]}^{\ell}\}_{\ell=0}^{\infty}$* can be defined, is an lfihd on $B$ of Jacobian type determined by $F$ such that $B^{D}=R[F]$ and has a slice.*

Proof.

(i) $\Longrightarrow$ (ii)  Since $F$ is extendable, there exists $s\in B$ such that $R[f_{1},\ldots,f_{n-1},s]=B$. Define the $R$-automorphism $\sigma:B\to B$ by $\sigma(x_{i})=f_{i}$ for $1\leq i\leq n-1$ and $\sigma(x_{n})=s$. We may assume that $\widetilde{\Delta}_{F}(s)=1$ and ${}^{\sigma}\widetilde{\Delta}_{F}=\partial_{x_{n}}$. It is clear that ${\rm Exp}(t\partial_{x_{n}})$ can be defined, hence ${\rm Exp}(t\cdot{}^{\sigma}\widetilde{\Delta}_{F})=\sigma^{-1}\circ{\rm Exp}(t\partial_{x_{n}})\circ\sigma$. This implies that ${\ell!}^{-1}{[\widetilde{\Delta}_{F}]}^{\ell}$ is defined at any $g\in B$ and $\ell\geq 1$. Set $D=\{{\ell!}^{-1}{[\widetilde{\Delta}_{F}]}^{\ell}\}_{\ell=0}^{\infty}$. Then $D$ is an lfihd on Jacobian type determined by $F$. It is clear that $B^{D}=R[F]$ and $s$ is a slice of $D$.

(ii) $\Longrightarrow$ (i)  Let $s\in B$ be a slice of $D$. By Proposition 1.1, $B=R[F][s]$, which implies that $F$ is extendable. ∎

3. Higher derivations of Jacobian type on $k[x,y]$

Let $k$ be a field of characteristic $p>0$. Through this section, we suppose that $k[x,y]\cong_{k}k^{[2]}$ is the polynomial ring in two variables over $k$.

By using Theorem 2.8, we have the following result. This is a generalization of [3, Theorem 3.2] in the case where the characteristic of the ground field is positive.

Theorem 3.1**.**

Let $f\in k[x,y]$. Then the following conditions are equivalent:**

1. (i)

$f$* is a variable.* 2. (ii)

$D=\{{\ell!}^{-1}{[\widetilde{\Delta}_{f}]}^{\ell}\}_{\ell=0}^{\infty}$* can be defined, is an lfihd on $k[x,y]$ of Jacobian type determined by $f$ such that $B^{D}=k[f]$ and has a slice.* 3. (iii)

$D=\{{\ell!}^{-1}{[\widetilde{\Delta}_{f}]}^{\ell}\}_{\ell=0}^{\infty}$* can be defined and is an lfihd on $k[x,y]$ of Jacobian type determined by $f$ such that $B^{D}=k[f]$.*

Proof.

(i) $\Longrightarrow$ (ii)  This implication follows from Theorem 2.8.

(ii) $\Longrightarrow$ (iii)  Obvious.

(iii) $\Longrightarrow$ (i)  Since the ring $k[f]$ is the kernel of the lfihd $D$, it follows from [5, Theorem 1] that $f$ is a variable. ∎

By using Theorem 3.1, we have the following. This is a generalization of [4, Corollary 4.6] in the case where the characteristic of the ground field is positive.

Corollary 3.2**.**

For $f\in k[x,y]\setminus k[x^{p},y^{p}]$, the following two conditions are equivalent:**

1. (i)

$f$* is univariate.* 2. (ii)

$D=\{{\ell!}^{-1}{[\widetilde{\Delta}_{f}]}^{\ell}\}_{\ell=0}^{\infty}$* can be defined and is an lfihd on $k[x,y]$ of Jacobian type determined by $f$.*

Proof.

(i) $\Longrightarrow$ (ii)  Since $f$ is univariate, there exists a variable $g\in k[x,y]$ such that $f\in k[g]$. Then $f=u(g)$ for some $u(t)\in k[t]\cong_{k}k^{[1]}$. Hence $\widetilde{\Delta}_{f}=u^{\prime}(g)\widetilde{\Delta}_{g}$, where $u^{\prime}(t)$ is the derivative of $u(t)$ with respect to $t$. By Theorem 3.1, $\delta=\{\ell^{-1}[\widetilde{\Delta}_{g}]^{\ell}\}_{\ell=0}^{\infty}$ is an lfihd on $k[x,y]$ of Jacobian type determined by $g$ with $k[x,y]^{\delta}=k[g]$. Since $u^{\prime}(g)\in k[x,y]^{\widetilde{\Delta}_{g}}$, we have $[\widetilde{\Delta}_{f}]^{\ell}=u^{\prime}(t)^{\ell}[\widetilde{\Delta}_{g}]^{\ell}$ for any $\ell\geq 1$. Therefore, ${\ell!}^{-1}[\widetilde{\Delta}_{f}]=u^{\prime}(t)^{\ell}{\ell!}^{-1}[\widetilde{\Delta}_{g}]^{\ell}$ is defined. Here, $D=\{{\ell!}^{-1}[\widetilde{\Delta}_{f}]^{\ell}\}$ is a well-defined iterative higher $R$-derivation of Jacobian type determined by $f$. Furthermore, since $\delta$ is locally finite, so is $D$.

(ii) $\Longrightarrow$ (i)  Since $D$ is locally finite and iterative, it follows from [5, Theorem 1] that $k[x,y]^{D}=k[g]$ for some variable $g\in k[x,y]$. Since $D$ is of Jacobian type, we have $f\in k[x,y]^{D}=k[g]$. This implies that $f$ is univariate. ∎

Remark 3.3**.**

In Theorems 2.8 and 3.1, we need to consider $\widetilde{\Delta}_{F}$, not ${\Delta}_{F}$.

For example, we consider $f=x-y^{3}\in\mathbb{F}_{3}[x,y]\cong_{\mathbb{F}_{3}}\mathbb{F}_{3}^{[2]}$ (see also Example 2.7). Then $\widetilde{\Delta}_{f}=3y^{2}\partial_{x}+\partial_{y}$ and $\Delta_{f}=\partial_{y}$. Clearly, both of $D_{1}:=\{{\ell!}^{-1}{[\widetilde{\Delta}_{f}]}^{\ell}\}_{\ell=0}^{\infty}$ and $D_{2}:=\{{\ell!}^{-1}{[{\Delta}_{f}]}^{\ell}\}_{\ell=0}^{\infty}$ are well-defined, locally finite and iterative. However, since $\varphi_{D_{2}}$ is defined by

[TABLE]

we have $\varphi_{D_{2}}(x-y^{3})=x-y^{3}-t^{3}$. Therefore $f\not\in\mathbb{F}_{3}[x,y]^{D_{2}}$, which implied that $D_{2}$ is not of Jacobian type determined by $f$. **