Introduction

Let

A
*derivation*

for any
*stable* if
*simple*. Even in the case of two variable polynomials, only a few examples of simple derivations are known (see, for instance, Brumatti et al. (^{ 2003}, Saraiva ^{ 2012}, Nowicki ^{ 2008}, Baltazar and Pan ^{ 2015}, Kour and Maloo ^{ 2013}, Lequain ^{ 2011})).

We denote by

Fix a derivation

We are interested in the following question proposed by I.Pan (see Baltazar (^{ 2014})):

**Conjecture 1**
*If
*

Initially, in Section 2, we prove Theorem 6, which shows that the conjecture is true for a family of derivations, namely Shamsuddin derivations. For this purpose, we use a theorem due to Shamsuddin (^{ 1977}) (see also ^{Nowicki 1994}, Theorem 13.2.1.) that gives a necessary and sufficient condition for a derivation to be extended to
^{ 2011}) in order to establish a conjecture about the Weyl algebra

In Section 3, to understand the isotropy of a simple derivation of
*dynamical degree* of a polynomial map and prove in Corollary 9 that for
^{ 1989})).

SHAMSUDDIN DERIVATIONS

The aim of this section is study the isotropy group of a Shamsuddin derivation in
^{ 1994}), there are numerous examples of these derivations and a criterion for determining the simplicity. Furthermore, Lequain (^{ 2008}) introduced an algorithm for determining whether a Shamsuddin derivation is simple. We begin with an example that shows that the isotropy of an arbitrary derivation can be quite complicated.

*Example 1*
*Let
*

*Since
*

1)

*Thus,*

*Then,
*

2)

*Analogously,*

*that is,
*

*Thus,
*

*with
*

*with
*

The following is a well known lemma.

**Lemma 2**
*Let
*

We also use the following result of Shamsuddin (^{ 1977}).

**Theorem 3**
*Let
*

*
*

*There exist no elements
*

*Proof* See (^{Nowicki 1994}, Theorem 13.2.1.) for a detailed proof.

A derivation
*Shamsuddin derivation* if

where

*Example 4*
*Let
*

*Writing
*. Thus, we know that

**Lemma 5**. (^{Nowicki 1994}, Proposition. 13.3.2) *Let
*

*Proof* If

One can determine the simplicity of the a Shamsuddin derivation according the polynomials
^{Nowicki 1994}, §13.3).

**Theorem 6**
*Let
*

*Proof* Let us denote

where

Then, by condition

with

By comparing the coefficients of

which can not occur by simplicity. More explicitly, Lemma 5 implies

By using condition

By the previous part, we can suppose that

By comparing the coefficients of

Then
^{Lequain 2008}, Lemma.2.6 and Theorem.3.2); thus, we obtain

Note that

Considering the independent term of

By (^{Nowicki 1994}, Proposition. 13.3.3), if

is a simple derivation. Furthermore, by Theorem 3, there exist no elements

This contradicts equation “eqrefeq1.1. Then,

ON THE ISOTROPY OF THE SIMPLE DERIVATIONS

The purpose of this section is to study the isotropy in the general case of a simple derivation. More precisely, we obtain results that reveal nice features of the elements of

In Baltazar and Pan (^{ 2015}), which was inspired by Brumatti et al. (^{ 2003}), the authors introduce and study a general notion of solution associated to a Noetherian differential

The following proposition has a geometrical flavour: it says that if an element in the isotropy of a simple derivation has fixed point, then it is the identity automorphism.

**Proposition 7**
*Let
*

*Proof* Let
^{Baltazar and Pan 2015, Definition 1}). We know that

In other words,
^{Baltazar and Pan 2015}, Theorem.7.(c)),

Lane (^{ 1975}) proved that every
^{ 1982}), Shamsuddin proved that this result does not extend to

In addition, since

must stabilize; thus, there exists a positive integer

Suppose that
^{Kaplansky 1974}, Theorem 5). If

we claim that
^{Atiyah and Macdonald 1969}, Prop.11.1.(ii)). Then,

Note that
^{Atiyah and Macdonald 1969}, Prop.11.1.(ii)): a contradiction. Thus, since

In the next corollary, we obtain consequences on the case of radical ideals.

**Corollary 8**
*Let
*

*Proof* Suppose that

Let
^{Farkas and Kra 1992}, Theorem Hunvitz, p.241), the number of elements in

In the rest of this section, we let

Consider a polynomial map
^{4}), Friedland and Milnor (^{ 1989}), Silverman (^{ 2012})) of

**Corollary 9**
*If
*

*Proof* Suppose
^{Friedland and Milnor 1989}, Theorem 3.1.),