Generalized Nowicki conjecture
Authors: Drensky Vesselin 
Published: February 2020

Let

B
B

be an integral domain over a field

K
K

of characteristic 0. The derivation

δ
\delta

of

B
[

Y
d

]
=
B
[

y
1

,
…
,

y
d

]

B[Y_d]=B[y_1,\ldots ,y_d]

is elementary if

δ
(
B
)
=
0

\delta (B)=0

and

δ
(

y
i

)
∈
B

\delta (y_i)\in B

,

i
=
1
,
…
,
d

i=1,\ldots ,d

. Then for

d
≥
2

d\geq 2

the elements

u

i
j

=
δ
(

y
i

)

y
j

−
δ
(

y
j

)

y
i

u_{ij}=\delta (y_i)y_j-\delta (y_j)y_i

,

1
≤
i
>
j
≤
d

1\leq i>j\leq d

, belong to the algebra

B
[

Y
d

]

δ

B[Y_d]^{\delta }

of constants of

δ
\delta

, and it is a natural question whether the

B
B

-algebra

B
[

Y
d

]

δ

B[Y_d]^{\delta }

is generated by all

u

i
j

u_{ij}

. In this paper we consider the special case of

B
=
K
[

X
d

]
=
K
[

x
1

,
…
,

x
d

]

B=K[X_d]=K[x_1,\ldots ,x_d]

. If

δ
(

y
i

)
=

x
i

\delta (y_i)=x_i

,

i
=
1
,
…
,
d

i=1,\ldots ,d

, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case

δ
(

y
i

)
=

x
i

n
i

\delta (y_i)=x_i^{n_i}

,

n
i

>
0

n_i>0

,

i
=
1
,
…
,
d

i=1,\ldots ,d

, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if

δ
(

y
i

)
=

f
i

(

x
i

)

\delta (y_i)=f_i(x_i)

, for any nonconstant polynomials

f
i

(

x
i

)

f_i(x_i)

,

i
=
1
,
…
,
d

i=1,\ldots ,d

, then

B
[

Y
d

]

δ

=
K
[

X
d

,

Y
d

]

δ

B[Y_d]^{\delta }=K[X_d,Y_d]^{\delta }

is generated by

X
d

X_d

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript d Baseline equals left-brace u Subscript i j Baseline equals f Subscript i Baseline left-parenthesis x Subscript i Baseline right-parenthesis y Subscript j Baseline minus y Subscript i Baseline f Subscript j Baseline left-parenthesis x Subscript j Baseline right-parenthesis bar 1 less-than-or-equal-to i greater-than j less-than-or-equal-to d right-brace">

U
d

=
{

u

i
j

=

f
i

(

x
i

)

y
j

−

y
i

f
j

(

x
j

)
∣
1
≤
i
>
j
≤
d
}

U_d=\{u_{ij}=f_i(x_i)y_j-y_if_j(x_j)\mid 1\leq i>j\leq d\}

. In the present paper we have found a presentation of the algebra
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-bracket upper X Subscript d Baseline comma upper Y Subscript d Baseline right-bracket Superscript delta Baseline equals upper K left-bracket upper X Subscript d Baseline comma upper U Subscript d Baseline bar upper R equals upper S equals 0 right-bracket comma d greater-than-or-equal-to 4 comma">

K
[

X
d

,

Y
d

]

δ

=
K
[

X
d

,

U
d

∣
R
=
S
=
0
]
,
d
≥
4
,

K[X_d,Y_d]^{\delta }=K[X_d,U_d\mid R=S=0],d\geq 4,

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals left-brace r left-parenthesis i comma j comma k comma l right-parenthesis bar 1 less-than-or-equal-to i greater-than j greater-than k greater-than l less-than-or-equal-to d right-brace comma upper S equals left-brace s left-parenthesis i comma j comma k right-parenthesis bar 1 less-than-or-equal-to i greater-than j greater-than k less-than-or-equal-to d right-brace comma">

R
=
{
r
(
i
,
j
,
k
,
l
)
∣
1
≤
i
>
j
>
k
>
l
≤
d
}
,
S
=
{
s
(
i
,
j
,
k
)
∣
1
≤
i
>
j
>
k
≤
d
}
,

R=\{r(i,j,k,l)\mid 1\leq i>j>k>l\leq d\}, S=\{s(i,j,k)\mid 1\leq i>j>k\leq d\},

\]
and a basis of

K
[

X
d

,

Y
d

]

δ

K[X_d,Y_d]^{\delta }

as a vector space. As a corollary we have shown that the defining relations

R
∪
S

R\cup S

form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras

K
[

X
d

,

Y
d

]

δ

K[X_d,Y_d]^{\delta }

,

d
>
4

d>4

, are also described.