Variable lemma for polynomial rings
Let \(k\) be a field of characteristic zero, and let \(k[x_1,\ldots,x_n]\) be a polynomial ring in \(n\) variables, where \(n\geq 3\) is an arbitrary positive integer. Assume that we have \(l\in\mathbb{N}\) algebraically independent polynomials \(F_1,\ldots,F_l\in k[x_1,\ldots,x_n]\) with \(2\leq l...
Збережено в:
| Дата: | 2026 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2026
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2428 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
| Завантажити файл: |  |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1870196963583983616 |
|---|---|
| author | Holik, Daria Karaś, Marek |
| author_facet | Holik, Daria Karaś, Marek |
| author_institution_txt_mv | [
{
"author": "Daria Holik",
"institution": "AGH University of Krakow\nFaculty of Applied Mathematics\nAl. A. Mickiewicza 30\n30-059 Kraków, Poland"
},
{
"author": "Marek Karaś",
"institution": "AGH University of Krakow\nFaculty of Applied Mathematics\nAl. A. Mickiewicza 30\n30-059 Kraków, Poland"
}
] |
| author_sort | Holik, Daria |
| baseUrl_str | https://admjournal.luguniv.edu.ua/index.php/adm/oai |
| collection | OJS |
| datestamp_date | 2026-07-08T07:55:33Z |
| description | Let \(k\) be a field of characteristic zero, and let \(k[x_1,\ldots,x_n]\) be a polynomial ring in \(n\) variables, where \(n\geq 3\) is an arbitrary positive integer. Assume that we have \(l\in\mathbb{N}\) algebraically independent polynomials \(F_1,\ldots,F_l\in k[x_1,\ldots,x_n]\) with \(2\leq l <n.\) In this paper, we prove that if linear parts of polynomials \(F_1,\ldots,F_l\) are linearly independent and depend only on variables \(x_1,\ldots,x_l\) and the polynomials \(F_1,\ldots,F_l\) meet some weighted-differential criteria then actually \(F_1,\ldots,F_l\in k[x_1,\ldots,x_l].\) |
| doi_str_mv | 10.12958/adm2428 |
| first_indexed | 2026-07-09T01:00:10Z |
| format | Article |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 41 (2026). Number 2, pp. 201–218
DOI:10.12958/adm2428
Variable lemma for polynomial rings
Daria Holik* and Marek Karaś
Communicated by I. Shestakov
Abstract. Let k be a field of characteristic zero, and let
k[x1, . . . , xn] be a polynomial ring in n variables, where n ≥ 3 is an
arbitrary positive integer. Assume that we have l ∈ N algebraically
independent polynomials F1, . . . , Fl ∈ k[x1, . . . , xn] with 2 ≤ l < n.
In this paper, we prove that if linear parts of polynomials F1, . . . , Fl
are linearly independent and depend only on variables x1, . . . , xl
and the polynomials F1, . . . , Fl meet some weighted-differential cri-
teria then actually F1, . . . , Fl ∈ k[x1, . . . , xl].
Introduction
In the landmarking paper [8] Shestakov and Umirbaev proved, in par-
ticular, that the following polynomial mapping, named Nagata automor-
phism
σ : C3 ∋ (x, y, z) 7→ (x+ 2y(y2 + xz)− z(y2 + xz)2,
y − z(y2 + xz), z) ∈ C3.
(1)
is so-called wild (polynomial) automorphism, that is, it is not a composi-
tion of linear/affine automorphisms and triangular automorphisms. Let
us recall that a polynomial automorphism F is called triangular if (after
*Corresponding author.
2020 Mathematics Subject Classification: Primary: 13N05, 14R10. Secon-
dary: 13B25, 13N15, 14L40.
Key words and phrases: polynomial automorphism, tame (wild) automor-
phism, (weighted) multidegree, general algebraic group.
https://doi.org/10.12958/adm2428
202 Variable lemma for polynomial rings
a possible permutation of variables and/or possible permutation of coor-
dinate functions F1, . . . , Fn of F ) the functions Fi have the following form
F1 = x1 + a with a ∈ k, F2 = x2 + P2 with P2 ∈ k[x1], . . . , Fi = xi + Pi
with Pi ∈ k[x1, . . . , xi−1], . . . , Fn = xn + Pn with Pn ∈ k[x1, . . . , xn−1].
The motivation of the research presented in this paper is the ap-
plication of the following so-called variable lemma (giving relationship
between the degree of the Poisson bracket and the number of variables)
Lemma 1 ([5, Lemma 3.20]). Let f, g ∈ k[x1, . . . , xn] be such that
f = x1 + f2 + . . .+ fr, g = x2 + g2 + . . .+ gs,
where fi, gi are homogeneous forms of degree i. If deg[f, g] = 2 then
f, g ∈ k[x1, x2], where
deg[f, g] = 2 + max
1≤i<j≤n
deg
(
∂f
∂xi
∂g
∂xj
− ∂f
∂xj
∂g
∂xi
)
in the studying the following conjecture concerning the existence of tame
automorphism with given degrees of its components.
Conjecture 1 ([3, Conj. 5.1]). For any prime number p ≥ 2 and d3 ≥
d2 ≥ p there exists a tame automorphism of k3 with multidegree equal to
(p, d2, d3) if and only if p|d2 or d3 ∈ pN+ d2N.
The conjecture, up to now, is proved for p = 2, 3, 5 (see [4, 5]). For
p = 5 the most difficult part was to show that there is no tame auto-
morphism F : k3 → k3 such that mdegF = (5, 6, 9), where mdegF =
(degF1, . . . ,degFm) if F = (F1, . . . , Fm) : kn → km is a polynomial
mapping and in proving this, Lemma 1 was used (for details see [5]).
The application of the weighted multidegrees in studying tameness
of polynomial automorphisms can be found in [1]. For example, among
other things, in [1] it was established that for w = (5, 4, 3) the w-multi-
degree of the Nagata automorphism, mdegw σ = (19, 11, 3) is the so-
called wild w-multidegree, i.e. there is no tame automorphism with the
same w-multidegree.
In the present paper we establish a collection of generalizations of the
above mentioned variable lemma (for more than two polynomials and for
weighted degrees; see Section 2). The authors believe that the results of
Section 2 can be used in the future in studying tameness in dimensions
higher than three.
It should be emphasized that the question about existence of wild
automorphism of kn for n ≥ 4 is still open. The situation is even worse:
D. Holik, M. Karaś 203
there is no tools for researching tameness/wildness in dimension n ≥ 4.
Perhaps as one of such tools can be regarded the result of Kuroda [6] and
as the second one the variable lemma presented in this paper (Theorem 1
below).
1. Differential forms and their degrees
In this section, for the convenience of the reader, we collect some neces-
sary facts about differential forms. At the beginning we recall that for
any polynomial H ∈ k[x1, . . . , xn], i.e. H =
∑
α=(α1,...,αn)∈Nn
aαx
α1
1 · · ·xαn
n ,
degH denotes the usual total degree of H that is
degH := max{α1 + · · ·+ αn | aα ̸= 0}
or degH := −∞ when H = 0. We remind that a homogeneous polyno-
mial is a polynomial whose nonzero terms all have the same degree.
Let A = k[x1, . . . , xn] be a polynomial ring and ΩA/k = A ·
{
dH | H ∈
k[x1, . . . , xn]
}
an A-module of differential forms of k[x1, . . . , xn] over k
(for definition and more information about differential forms we refer to
[7, Chapter 5.3] or [2, Chapter 2.8]). One can write ΩA/k =
n⊕
i=1
Adxi,
and then
∧pΩA/k =
⊕
1⩽i1<...<ip⩽n
Adxi1 ∧ . . . ∧ dxip , where the symbol∧pΩA/k denotes the p-th exterior power of the A-module ΩA/k. Arbitrary
differential form ω ∈
∧pΩA/k, which is said to be of order p, one can
uniquely express as
ω =
∑
1⩽i1<...<ip⩽n
φi1,...,ipdxi1 ∧ . . . ∧ dxip ,
where φi1,...,ip ∈ A for all 1 ≤ i1 < . . . < ip ≤ n.
We define standard total degree of a differential form ω ∈
∧pΩA/k
by
degω := max
1⩽i1<...<ip⩽n
{degφi1,...,ip + p}.
Using the fact that the field k is of characteristic zero and dH=
n∑
i=1
∂H
∂xi
dxi,
we get the following equality
deg dH = max
i=1,...,n
{
deg
∂H
∂xi
+ 1
}
= degH
204 Variable lemma for polynomial rings
for each H ∈ k[x1, . . . , xn] \ k.
It is natural to say that a differential form ω =
∑
1⩽i1<...<ip⩽n
φi1,...,ip
·dxi1∧. . .∧dxip is homogeneous of degree d if for all 1 ⩽ i1 < . . . < ip ⩽ n
with φi1,...,ip ̸= 0, the coefficient φi1,...,ip is a homogeneous polynomial
such that degφi1,...,ip + p is equal to d.
Now we can formulate the following
Lemma 2. If H1, . . . ,Hp ∈ k[x1, . . . , xn] are nonzero, homogeneous poly-
nomials in n variables x1, . . . , xn, then the differential form dH1 ∧ . . . ∧
dHp is homogeneous of degree
deg(dH1 ∧ . . . ∧ dHp)
=
{
degH1 + · · ·+ degHp if dH1 ∧ . . . ∧ dHp ̸= 0,
−∞ if dH1 ∧ . . . ∧ dHp = 0.
(2)
The proof of the above lemma is a consequence of the definition of
the degree of a differential form and the straightforward calculations. We
omit it, since below we give a generalization of this lemma with a detailed
proof (see Lemma 3).
Let Γ be any totally ordered additive group, and w = (w1, . . . , wn) ∈
Γn with wi ⩾ 0, i = 1, . . . , n. The polynomial ring k[x1, . . . , xn] in
n variables over a field k of characteristic zero with n ∈ N can be
equipped with the w-weighted grading structure as follows k[x1, . . . , xn]
=
⊕
γ∈Γ
k[x1, . . . , xn]γ , where k[x1, . . . , xn]γ , for any γ ∈ Γ, denotes k-vec-
tor space spanned by xα1
1 · · ·xαn
n for α1, . . . , αn ∈ N with
n∑
i=1
αiwi = γ.
Now we can define w-degree of the polynomial H =
∑
γ∈Γ
Hγ , where for
any γ ∈ Γ, Hγ ∈ k[x1, . . . , xn]γ is the homogeneous component of degree
γ of the polynomial H, in the following way
degw H = max{γ ∈ Γ | Hγ ̸= 0}
or degw H = −∞ while H = 0. It should be mentioned that if Γ = Z
and w = (1, . . . , 1), then w-degree is equal to the usual total degree. It
is easy to see that for H =
∑
α=(α1,...,αn)∈Nn
aαx
α1
1 · · ·xαn
n and any γ ∈ Γ,
we have Hγ =
∑
α1w1+···+αnwn=γ
aαx
α1
1 · · ·xαn
n .
D. Holik, M. Karaś 205
If for some γ ∈ Γ the following equality holds H = Hγ , then H
is called w-homogeneous. Analogously to the usual total degree of the
differential form ω ∈
∧pΩA/k, we define the w-degree of ω as follows
degw ω := max
1⩽i1<...<ip⩽n
{degw φi1,...,ip + wi1 + · · ·+ wip}.
As in the case of polynomials, if Γ = Z and w = (1, . . . , 1) then the
w-degree of differential form is equal to its usual total degree, i.e degw ω
= degω for all ω ∈
∧pΩA/k. It is natural to say that a differential form
ω =
∑
1⩽i1<...<ip⩽n
φi1,...,ipdxi1 ∧ . . . ∧ dxip is w-homogeneous of w-degree
γ if for every 1 ⩽ i1 < . . . < ip ⩽ n with φi1,...,ip ̸= 0, the polynomial
φi1,...,ip is w-homogeneous and degw φi1,...,ip + wi1 + · · ·+ wip = γ.
The following lemma will be used in the proofs of the main results,
in the next section.
Lemma 3. Let w = (w1, . . . , wn) ∈ Γn be any n-tuple of elements of the
group Γ with wi ≥ 0 for i = 1, . . . , n. If H1, . . . ,Hp ∈ k[x1, . . . , xn] are
nonzero, w-homogeneous polynomials in n variables x1, . . . , xn, then the
differential form dH1 ∧ . . . ∧ dHp is w-homogeneous of w-degree
degw(dH1 ∧ . . . ∧ dHp)
=
{
degw H1 + · · ·+ degw Hp if dH1 ∧ . . . ∧ dHp ̸= 0,
−∞ if dH1 ∧ . . . ∧ dHp = 0.
(3)
Proof. Since, for any i ∈ {1, . . . , p}, we have dHi =
n∑
j=1
∂Hi
∂xj
dxj , it follows
from the multilinearity of the wedge product that
dH1 ∧ . . . ∧ dHp =
( n∑
j1=1
∂H1
∂xj1
dxj1
)
∧ · · · ∧
( n∑
jp=1
∂Hp
∂xjp
dxjp
)
=
n∑
j1=1
. . .
n∑
jp=1
∂H1
∂xj1
· · · ∂Hp
∂xjp
dxj1 ∧ . . . ∧ dxjp .
(4)
Because the wedge product is antisymmetric, the above sum can be writ-
ten as
dH1 ∧ . . . ∧ dHp
=
∑
1≤j1<...<jp≤n
(∑
σ∈Sp
sgnσ
∂H1
∂xjσ(1)
· · · ∂Hp
∂xjσ(p)
)
dxj1 ∧ . . . ∧ dxjp ,
(5)
206 Variable lemma for polynomial rings
where Sp denotes the symmetric group of the set {1, . . . , p} and the sgnσ
is the sign of the permutation σ. Thus, by the definition of w-degree of
a differential form and by (5), we have
degw(dH1 ∧ . . . ∧ dHp)
= max
1≤j1<...<jp≤n
{
degw
(∑
σ∈Sp
sgnσ
∂H1
∂xjσ(1)
· · · ∂Hp
∂xjσ(p)
)
+ wj1+· · ·+wjp
}
.
(6)
Using the fact that k is of characteristic zero, we obtain that the
polynomial ∂Hi
∂xjσ(m)
is w-homogeneous and degw
∂Hi
∂xjσ(m)
= degw Hi −
wjσ(m)
iff ∂Hi
∂xjσ(m)
̸= 0, for any m = 1, . . . , p, any σ ∈ Sp, and any
1 ≤ j1 < . . . < jp ≤ n. Consequently, any summand of the expression∑
σ∈Sp
sgnσ ∂H1
∂xjσ(1)
· · · ∂Hp
∂xjσ(p)
is a w-homogeneous polynomial which has
w-degree equal to degw H1 − wjσ(1)
+ · · · + degw Hp − wjσ(p)
or it is
equal to zero. Taking into consideration that σ is bijective function,
the above w-degree we can write as follows degw H1 + · · · + degw Hp −
wj1 − · · · − wjp . Finally, for any 1 ≤ j1 < . . . < jp ≤ n, the polynomial∑
σ∈Sp
sgnσ ∂H1
∂xjσ(1)
· · · ∂Hp
∂xjσ(p)
is w-homogeneous, and its w-degree equals
degw H1 + · · ·+ degw Hp −wj1 − · · · −wjp . From the above calculations
and (6) we obtain the assertion.
2. Main results – variable lemmas
In this section we present main results of the paper, that is Theorems 1
and 2. Both of them can be named variable lemma because they assert
that some polynomials that a priori depend on the whole set of variables
x1, . . . , xn under some assumptions actually appear to be dependent only
on some proper subset of variables. One can also be interested in Corol-
laries 1 and 2 which can be considered, respectively, as a generalization
of Theorems 2 and 1.
Before we give the results let us formulate the following two condi-
tions:
(A)
Γ is an ordered additive group such that every finitely
generated subgroup is embedded in a subgroup of Γ which
is isomorphic to (Z,+), in the category of ordered groups.
D. Holik, M. Karaś 207
(B)
Γ is an ordered additive group and w = (w1, . . . , wn) ∈ Γn
is such that the subgroup Zw1+· · ·+Zwn ⊂ Γ is embedded
in a subgroup of Γ which is isomorphic to (Z,+), in the
category of ordered groups.
One can easily see that condition (A) implies (B), and moreover Γ = Z
satisfies condition (A). In the next section we show example of totally
ordered group which does not satisfy condition (A) as well an example
of Γ and w = (w1, . . . , wn) ∈ Γn such that the condition (B) is not
satisfied. The example of group which is not isomorphic to (Z,+) but
satisfies the condition (A) is also presented there. As well we present, in
Section 3, examples showing that assumptions of the following theorem
are essential.
Theorem 1. Assume that Γ is a totally ordered additive group and that
w = (w1, . . . , wn) ∈ Γn is such that wi > 0 for i = 1, . . . , n, and
max{w1, . . . , wl} ≤ min{wl+1, . . . , wn}. (7)
Assume also that at least one of the conditions (A) or (B) is satisfied
(in particular, this is the case for Γ = Z). Let F1, . . . , Fl ∈ k[x1, . . . , xn],
where 2 ≤ l < n, be polynomials in n variables x1, . . . , xn such that
their linear parts are linearly independent and depend only on variables
x1, . . . , xl. If the w-degree of the differential form dF1 ∧ . . . ∧ dFl is the
smallest possible, i.e.
degw(dF1 ∧ . . . ∧ dFl) = w1 + · · ·+ wl, (8)
then we have F1, . . . , Fl ∈ k[x1, . . . , xl] ⊊ k[x1, . . . , xn].
Let us notice that the minimality of degw(dF1∧ . . .∧dFl), in the con-
text of Theorem 1, is equivalent to equality (8). Indeed, since
∧l ΩA/k =⊕
1⩽j1<...<jl⩽n
Adxj1∧. . .∧dxjl , it follows that for any nonzero ω ∈
∧l ΩA/k,
we have
degw ω ≥ min{wj1 + · · ·+ wjl | 1 ≤ j1 < . . . < jl ≤ n}.
When inequality (7) is satisfied, then we have
min{wj1 + · · ·+ wjl | 1 ≤ j1 < . . . < jl ≤ n} = w1 + · · ·+ wl,
and in the situation like in Corollary 2 we have
min{wj1 + · · ·+ wjl | 1 ≤ j1 < . . . < jl ≤ n}
= min{wj1 + · · ·+ wjl | 1 ≤ j1 < . . . < jl ≤ r}.
208 Variable lemma for polynomial rings
The proof of the above theorem will be given at the end of this section,
but now we present following particular/non-weighted version of it, i.e.
when Γ = Z and w = (1, . . . , 1). One can notice that if (w1, . . . , wn) =
(1, . . . , 1), then the assumption (7) is obviously satisfied.
Theorem 2. Let F1, . . . , Fl ∈ k[x1, . . . , xn], where 2 ≤ l < n, be poly-
nomials in n variables x1, . . . , xn such that their linear parts are linearly
independent and depend only on variables x1, . . . , xl.
If the degree of the differential form dF1 ∧ . . . ∧ dFl is the smallest
possible, i.e.
deg(dF1 ∧ . . . ∧ dFl) = l, (9)
then the polynomials F1, . . . , Fl involve only variables x1, . . . , xl. In other
words F1, . . . , Fl ∈ k[x1, . . . , xl] ⊊ k[x1, . . . , xn].
Before we give the proof of the above theorem let us explain why we
present it, besides of the fact that it is a particular version of Theorem 1.
The reason is the following. The proof of Theorem 1 is based of Propo-
sition 1, and the proof of Proposition 1 is a modification of the proof of
Theorem 2. Moreover, the situation of Theorem 2 makes that the idea
of the proof is much easier to present.
Proof. First of all we show that, without loss of generality, we can assume
that the polynomials F1, . . . , Fl have the form
Fi = xi + Fi,2 + · · ·+ Fi,di for i = 1, . . . , l, (10)
where Fi,j ∈ k[x1, . . . , xn] is a homogeneous component of degree j of
the polynomial Fi. Of course, one can notice that we can assume that
the constant terms of the polynomials F1, . . . , Fl are zeros. Let us denote
the linear parts of Fi by Li ∈ k[x1, . . . , xl] for i = 1, . . . , l, and consider
the mapping L−1, where L = (L1, . . . , Ll) : kl → kl. Now, by hi =
ai,1x1+ · · ·+ai,lxl, for i = 1, . . . , l, we denote the coordinate functions of
L−1. Then, det(ai,j) ̸= 0 and, as one can calculate using multilinearity
and antisymmetricity of the wedge product, that
dh1(F1, . . . , Fl) ∧ . . . ∧ dhl(F1, . . . , Fl) = det(ai,j)dF1 ∧ . . . ∧ dFl. (11)
This gives that the degrees of the differential forms dh1(F1, . . . , Fl)∧ . . .∧
dhl(F1, . . . , Fl) and dF1∧ . . .∧dFl are the same. One can also notice that
the polynomials h1(F1, . . . , Fl), . . . , hl(F1, . . . , Fl) have the desired form
and that these polynomials do not depend on variables xl+1, . . . , xn if
D. Holik, M. Karaś 209
and only if the same holds for the polynomials F1, . . . , Fl. Thus, up to
the end of the proof we assume that (10) holds, in other words, we have
Fi,1 = xi and Fi,0 = 0 for i = 1, . . . , l.
Now, from multilinearity of the wedge product and linearity of the
operator d, we have
dF1 ∧ . . . ∧ dFl =
( d1∑
j1=1
dF1,j1
)
∧ . . . ∧
( dl∑
jl=1
dFl,jl
)
=
d1∑
j1=1
· · ·
dl∑
jl=1
dF1,j1 ∧ . . . ∧ dFl,jl
=
d1+···+dl∑
d=0
∑
j1+···+jl=d
dF1,j1 ∧ . . . ∧ dFl,jl .
(12)
Observe that by Lemma 2 (or Lemma 3 for the weighted case) we have
(dFi)j = d(Fi,j). Thus we can omit brackets in formulas like above. By
Lemma 2, one can see that for any d = 0, . . . , d1 + · · · + dl, the sum∑
j1+···+jl=d
dF1,j1 ∧ . . . ∧ dFl,jl is the homogeneous component of the de-
gree d of the differential form dF1 ∧ . . . ∧ dFl. In particular, based on
the observation from the beginning of the proof, we see that the homo-
geneous components of the differential form dF1 ∧ . . . ∧ dFl of degree
d = 0, 1, . . . , l − 1 are equal to zero and the homogeneous component of
degree l is
(dF1 ∧ . . . ∧ dFl)l = dx1 ∧ . . . ∧ dxl ̸= 0. (13)
Notice that
(dF1 ∧ . . . ∧ dFl)l+1 = dF1,1 ∧ . . . ∧ dFl−1,1 ∧ dFl,2
+ dF1,1 ∧ . . . ∧ dFl−2,1 ∧ dFl−1,2 ∧ dFl,1
...
+ dF1,2 ∧ dF2,1 ∧ . . . ∧ dFl,1
= dx1 ∧ . . . ∧ dxl−1 ∧
( n∑
j=1
∂Fl,2
∂xj
dxj
)
+ dx1 ∧ . . . ∧ dxl−2 ∧
( n∑
j=1
∂Fl−1,2
∂xj
dxj
)
∧ dxl
...
+
( n∑
j=1
∂F1,2
∂xj
dxj
)
∧ dx2 ∧ . . . ∧ dxl.
210 Variable lemma for polynomial rings
Now, consider arbitrary j > l and notice that the only summand of
the above sum involving dx1 ∧ . . . ∧ dxl−1 ∧ dxj is the following one
∂Fl,2
∂xj
dx1∧ . . .∧dxl−1∧dxj . Since (dF1 ∧ . . . ∧ dFl)l+1 = 0, it follows that
∂Fl,2
∂xj
= 0 for all j > l. In other words the polynomial Fl,2 does not involve
variables xl+1, . . . , xn.
Similarly, one can notice that for j > l and for any i = 1, . . . , l − 1,
the only summand involving dx1 ∧ . . .∧ dxi−1 ∧ dxi+1 ∧ . . .∧ dxl ∧ dxj in
the above sum is the following one (−1)l−i ∂Fi,2
∂xj
dx1∧ . . .∧dxi−1∧dxi+1∧
. . .∧ dxl ∧ dxj . For the similar reason as above this means that
∂Fi,2
∂xj
= 0
and so that Fi,2 does not involve xj for any j > l and any i = 1, . . . , l−1.
Thus, we have proved that the homogeneous components of degree 2 of
polynomials F1, . . . , Fl do not involve variables xl+1, . . . , xn.
Now, assume that we have proved that the homogeneous components
of degree d = 2, . . . , s of the polynomials F1, . . . , Fl do not involve vari-
ables xl+1, . . . , xn. We will show that this is also true for homogeneous
components of degree s + 1. To do this consider the homogeneous com-
ponent of degree s + l of the differential form dF1 ∧ . . . ∧ dFl. One can
observe that
(dF1 ∧ . . . ∧ dFl)l+s
=
∑
j1+···+jl=l+s
j1,...,jl≤s
dF1,j1 ∧ . . . ∧ dFl,jl (14)
+
l∑
i=1
dF1,1 ∧ . . . ∧ dFi−1,1 ∧ dFi,s+1 ∧ dFi+1,1 ∧ . . . ∧ dFl,1. (15)
Take any j > l. By induction hypothesis, one can easily see that the sum
(14) does not involve dxj . On the other hand one can see that in the sum
(15) the only summand involving (dx1 ∧ . . . ∧ dxi−1 ∧ dxi+1 ∧ . . . ∧ dxl)∧
dxj is the following one
(−1)l−i∂Fi,s+1
∂xj
(dx1 ∧ . . . ∧ dxi−1 ∧ dxi+1 ∧ . . . ∧ dxl) ∧ dxj . (16)
From this, as before, it follows that Fi,s+1 does not involve xj for any
j > l and any i = 1, . . . , l. Now, by induction (with respect to s) we
obtain the thesis.
The result of the above theorem can be generalized as follows.
D. Holik, M. Karaś 211
Corollary 1. Let F1, . . . , Fl ∈ k[x1, . . . , xn], where 2 ≤ l < n, be poly-
nomials in n variables x1, . . . , xn such that their linear parts are linearly
independent and depend only on variables x1, . . . , xr for some l ≤ r < n.
If the degree of the differential form dF1 ∧ . . . ∧ dFl is the smallest
possible, i.e.
deg(dF1 ∧ . . . ∧ dFl) = l, (17)
then the polynomials F1, . . . , Fl involve only variables x1, . . . , xr, i.e. F1,
. . . , Fl ∈ k[x1, . . . , xr] ⊊ k[x1, . . . , xn].
Proof. Let us denote the linear parts of Fi by Li ∈ k[x1, . . . , xr] for
i = 1, . . . , l. There are linear forms Ll+1, . . . , Lr ∈ k[x1, . . . , xr] such that
L1, . . . , Ll, Ll+1, . . . , Lr are linearly independent. Now, one can check
that polynomials F1, . . . , Fl, Ll+1, . . . , Lr satisfy the assumptions of The-
orem 2 with l replaced by r. Thus, by Theorem 2, we obtain the the-
sis.
Similar arguments as in the proof of Corollary 1, but based on The-
orem 1, give the following result.
Corollary 2. Assume that Γ is a totally ordered additive group and that
w = (w1, . . . , wn) ∈ Γn is such that wi > 0 for i = 1, . . . , n, and
max{w1, . . . , wr} ≤ min{wr+1, . . . , wn}. (18)
Assume also that at least one of the conditions (A) or (B) is satisfied.
Let F1, . . . , Fl ∈ k[x1, . . . , xn], where 2 ≤ l < n be polynomials in n
variables x1, . . . , xn such that their linear parts are linearly independent
and depend only on variables x1, . . . , xr, where l ≤ r < n. If the w-degree
of the differential form dF1 ∧ . . . ∧ dFl is the smallest possible then we
have F1, . . . , Fl ∈ k[x1, . . . , xr] ⊊ k[x1, . . . , xn].
The following technical proposition gives a weighted version of Theo-
rem 2 (with some additional assumptions), and will be used in the proof
of Theorem 1. Because in Theorem 1 we make an assumption (7), which
is absent in the proposition below, then the following result can be in-
teresting on its own right.
Proposition 1. Assume that w = (w1, . . . , wn) ∈ Zn is such that wi > 0
for i = 1, . . . , n. Let F1, . . . , Fl ∈ k[x1, . . . , xn], where 2 ≤ l < n be
polynomials in n variables x1, . . . , xn such that
Fi = xi + Fi,wi+1 + · · ·+ Fi,di for i = 1, . . . , l, (19)
212 Variable lemma for polynomial rings
where Fi,j ∈ k[x1, . . . , xn] is a w-homogeneous component of w-degree j
of the polynomial Fi. If the w-degree of the differential form dF1∧. . .∧dFl
is the smallest possible, i.e.
degw(dF1 ∧ . . . ∧ dFl) = w1 + · · ·+ wl, (20)
then the polynomials F1, . . . , Fl involve only variables x1, . . . , xl, i.e. F1,
. . . , Fl ∈ k[x1, . . . , xl] ⊊ k[x1, . . . , xn].
Proof. The proof is the appropriate modification and extension of the
proof of Theorem 2. First, observe that by the assumption, we have
Fi,0 = Fi,1 = . . . = Fi,wi−1 = 0 (21)
and
Fi,wi = xi (22)
for all i = 1, . . . , l.
Secondly, similarly as in the proof of Theorem 2, we have
dF1 ∧ . . . ∧ dFl =
d1+···+dl∑
d=0
∑
j1+···+jl=d
dF1,j1 ∧ . . . ∧ dFl,jl , (23)
where, this time, Fi,j means the w-homogeneous component of w-degree
j of the polynmial Fi.
Now, by Lemma 3, we have that for any number d ∈ N the sum∑
j1+···+jl=d
dF1,j1 ∧ . . . ∧ dFl,jl is the w-homogeneous component of
w-degree d of the differential form dF1 ∧ . . .∧ dFl. Without loss of gene-
rality, we can assume that w1 ≤ . . . ≤ wl.
Let us take any d ∈ {0, . . . , w1 + · · · + wl − 1}. Since for any non-
negative integers j1, . . . , jl with j1 + · · · + jl = d at least one of them
satisfies ji < wi (for some i ∈ {1, . . . , l}), we see that the w-homogeneous
components of the w-degree d = 0, 1, . . . , w1 + · · · + wl − 1, of the dif-
ferential form dF1 ∧ . . . ∧ dFl, are equal to zero. One can also notice
that the w-homogeneous component of the w-degree w1 + · · · + wl of
dF1 ∧ . . . ∧ dFl is
(dF1 ∧ . . . ∧ dFl)w1+···+wl
= dx1 ∧ . . . ∧ dxl ̸= 0. (24)
D. Holik, M. Karaś 213
To imitate the proof of Theorem 2, now we consider w-homogeneous
component of w-degree w1 + · · ·+ wl + 1. One can notice that
(dF1 ∧ . . . ∧ dFl)w1+···+wl+1
= dF1,w1 ∧ . . . ∧ dFl−1,wl−1
∧ dFl,wl+1
+ dF1,w1 ∧ . . . ∧ dFl−2,wl−2
∧ dFl−1,wl−1+1 ∧ dFl,wl
...
+ dF1,w1+1 ∧ dF2,w2 ∧ . . . ∧ dFl,wl
.
Similar analysis, as in the proof of Theorem 2, of the summands involving
dx1∧. . .∧dxl−1∧dxj , dx1∧. . .∧dxl−2∧dxl∧dxj , . . . , dx2∧. . .∧dxl∧dxj
with j > l that appear in the above sum gives us that for any i = 1, . . . , l
and for any j > l the polynomial Fi,wi+1 does not involve xj .
The induction step presented in the proof of Theorem 2 needs follo-
wing modifications. Assume that we have proved that thew-homogeneous
components of w-degree di = wi + 1, . . . , wi + s with s ≥ 1 and for all
i = 1, . . . , l, do not involve xj for all j > l. Similarly, as in the proof
of Theorem 2, one can notice that (dF1 ∧ . . . ∧ dFl)w1+···+wl+s+1 can be
decomposed into two sums (compare (14) and (15)). One of them with
components such that j1 ≤ w1 + s, . . . , jl ≤ wl + s. This sum does not
involve dxj for j > l by induction hypothesis. The second sum have the
following form (compare (15))
l∑
i=1
dF1,w1 ∧ . . .∧dFi−1,wi−1 ∧dFi,wi+s+1∧dFi+1,wi+1 ∧ . . .∧dFl,wl
. (25)
By the similar arguments, as in the proof of Theorem 2, we obtain that for
all i = 1, . . . , l the polynomial Fi,wi+s+1 does not involve xj for all j > l.
This completes the induction step and the proof of the proposition.
Now we are in a position to prove Theorem 1.
Proof of Theorem 1. Since Γ and w = (w1, . . . , wn) ∈ Γn satisfy condi-
tion (B), there is subgroup Γ0 ⊂ Γ isomorphic to (Z,+) (in the category
of ordered groups) such that Zw1 + · · · + Zwn ⊂ Γ0. All considerations
will be done inside Γ0, because any w-degree (of polynomial or differen-
tial form) is an element of Nw1 + · · · + Nwn ⊂ Zw1 + · · · + Zwn. Thus
without loss of generality we can assume Γ = Z.
214 Variable lemma for polynomial rings
As in the proof of Theorem 2 we can assume that the polynomials
F1, . . . , Fl have the form given by (10), with respect to the usual de-
gree. By inequality (7), permutating if necessery variables in the two
sets {x1, . . . , xl} and {xl+1, . . . , xn} separately, we can assume, without
loss of generality, that
w1 ≤ . . . ≤ wl ≤ wl+1 ≤ . . . ≤ wn. (26)
Let
Fi = Fi,1 + · · ·+ Fi,wi−1 +
(
xi + F̃i,wi
)
+ Fi,wi+1 + · · ·+ Fi,di , (27)
for i=1, . . . , l, be the decomposition of the polynomial Fi into the
w-homogeneous components. In particular, xi+F̃i,wi is thew-homogene-
ous component of w-degree wi of Fi, while F̃i,wi is the w-homogeneous
component of w-degree wi of F̃i = Fi − xi.
The idea of the proof is to construct an automorphism Φ of kn of the
form
Φ = Φ̃× idkn−l , Φ(0) = 0, (28)
where Φ̃ is an automorphism of kl, and such that the polynomials F̂1,
. . . , F̂l satisfy the assumptions of Proposition 1, where (F̂1, . . . , F̂n) =
Φ ◦ (F1, . . . , Fl, xl+1, . . . , xn) : kn → kn. In this situation, of course, we
have F̂l+1 = xl+1, . . . , F̂n = xn. Then, by Proposition 1 one obtain that
the polynomials F̂1, . . . , F̂l do not involve variables xl+1, . . . , xn. Since
(F1, . . . , Fl, xl+1, . . . , xn) = Φ−1 ◦ (F̂1, . . . , F̂l, xl+1, . . . , xn) and because
of the form of the automorphism Φ, it follows that also F1, . . . , Fl do
not involve variables xl+1, . . . , xn. Thus, to the end of the proof we will
construct a desired polynomial automorphism.
First, notice that F1 already has the form as in the assumptions of
Proposition 1. Indeed, we have w1 ≤ w2 ≤ . . . ≤ wn and wi > 0 for
all i. Thus, for any monomial m of usual degree greater than 1, we have
degw m ≥ 2w1 > w1, but only such monomials appear in the polynomial
F1 − x1.
The construction of Φ will be done by induction. Assume that we
have already constructed an automorphism Φu = Φ̃u × idkn−u , where
Φ̃u is an automorphism of ku, for some u ∈ {1, . . . , l}, such that for
(F̂
(u)
1 , . . . , F̂
(u)
n ) = Φu ◦ (F1, . . . , Fl, xl+1, . . . , xn), we have
F̂
(u)
i = xi + higher w-degree components for i = 1, . . . , u (29)
D. Holik, M. Karaś 215
and
dF̂
(u)
1 ∧ . . . ∧ dF̂
(u)
l = dF1 ∧ . . . ∧ dFl. (30)
Let us notice that for u = 1, we can take Φu = idkn .
Now, let a1 be the w-homogeneous component of F̂
(u)
u+1 − xu+1 of
the minimal w-degree. If degw a1 > wu+1, then F̂
(u)
u+1 already has the
desired form, and so we can take Φu+1 = Φu. Thus, without loss of
generality, we can assume that degw a1 ≤ wu+1. Since all monomi-
als appearing in F̂
(u)
u+1 − xu+1 have usual degree greater than 1, it fol-
lows that a1 ∈ k[x1, . . . , xu]. Moreover, one can easily notice that the
w-homogeneous component of a1(F̂
(u)
1 , . . . , F̂
(u)
u ) of the minimalw-degree
is equal to a1. Now, let a2 be the w-homogeneous component of F̂
(u)
u+1 −
xu+1−a1(F̂
(u)
1 , . . . , F̂
(u)
u ) of the minimal w-degree. By the above consid-
erations, we have degw a2 > degw a1. If degw a2 ≤ wu+1, then as before
a2 ∈ k[x1, . . . , xu] and, also as before, the w-homogeneous component
of a2(F̂
(u)
1 , . . . , F̂
(u)
u ) of the minimal w-degree is equal to a2. In a simi-
lar way we construct w-homogeneous polynomials a1, . . . , as, as+1 until
degw as ≤ wu+1 and degw as+1 > wu+1. By the construction, we obtain
that the polynomial F̂
(u)
u+1 −
s∑
i=1
ai(F̂
(u)
1 , . . . , F̂
(u)
u ) has the desired form.
Moreover, by the fact that wedge product is multilinear and antisym-
metric, one can notice that
dF̂
(u)
1 ∧ . . . ∧ dF̂ (u)
u
∧ d
(
F̂
(u)
u+1 −
s∑
i=1
ai(F̂
(u)
1 , . . . , F̂ (u)
u )
)
∧ dF̂
(u)
u+2 ∧ . . . ∧ dF̂
(u)
l
= dF̂
(u)
1 ∧ . . . ∧ dF̂ (u)
u ∧ dF̂
(u)
u+1 ∧ dF̂
(u)
u+2 ∧ . . . ∧ dF̂
(u)
l
= dF1 ∧ . . . ∧ dFl.
(31)
Summarizing, we can take Φu+1 = φu+1 ◦ Φu, where the automor-
phism φu+1 is given by
φu+1 : k
n ∋(x1, . . . , xn) 7→
(x1, . . . , xu, xu+1 −
s∑
i=1
ai, xu+2, . . . , xn) ∈ kn.
(32)
This completes the proof of the induction step, and the existence of the
desired automorphism Φ = Φl.
216 Variable lemma for polynomial rings
3. Examples
In this section we show by giving examples, that not all totally ordered
groups satisfy condition (A) and (B), as well as that assumptions of
Theorem 1 are essential. We start with the following lemma.
Lemma 4. Assume that Γ is a totally ordered additive group and that
(w1, . . . , wn) ∈ Γn is such that the condition (B) is satisfied. Then
1. The set (Zw1+ · · ·+Zwn)∩Γ>0 has minimal element, where Γ>0 =
{γ ∈ Γ : γ > 0}.
2. For any a, b ∈ (Zw1 + · · ·+ Zwn) ∩ Γ>0 there is a natural number
k such that ka ≥ b.
Proof. Let Γ0 ⊂ Γ be a subgroup that is isomorphic to (Z,+), in the
category of ordered groups, such that Zw1 + · · · + Zwn ⊂ Γ0. To see
assertion (1) observe that (Zw1+ · · ·+Zwn)∩Γ>0 = (Zw1+ · · ·+Zwn)∩
(Γ0)>0. The last intersection can be viewed as a subset of Z>0. Thus one
can use the fact that any nonempty subset of Z>0 has a minimal element.
Assertion (2) is a consequence of the fact that we can assume, by the
above considerations, that a, b ∈ (Zw1 + · · ·+ Zwn) ∩ Γ>0 ⊂ Z>0.
Now, using the above lemma, we can present the following two examp-
les of Γ and w ∈ Γn which does not satisfy conditions (A) and (B).
Example 1. Consider the following totally ordered additive group Γ =
(R,+) and w = (w1, w2) ∈ R2, where w1 = 1 and w2 ∈ R \ Q. By the
ergodic theorem [9, Theorem 0.14(i)] for any ε > 0 there exist k ∈ N
such that 0 < kw2 − ⌊kw2⌋ < ε. Indeed, we can consider the subgroup
G = {e2πikw2 | k ∈ Z} of the unit circle K, with addition given by
K×K ∋ (e2πiα1 , e2πiα2) 7→ e2πi(α1+α2) ∈ K. In this situation G is infinite.
Moreover, one can check that the closureH of any subgroupH ofK is the
closed subgroup ofK. Thus, using [9, Theorem 0.14(i)] we obtain that the
group G is dense inK, and this implies that the set {kw2−⌊kw2⌋ | k ∈ Z}
is dense in [0, 1]. Taking l = −⌊kw2⌋ we have kw2−⌊kw2⌋ = lw1+kw2 ∈
Zw1 +Zw2. This shows that the set (Zw1 +Zw2)∩R>0 does not have a
minimal element. Thus, by Lemma 4(1) Γ and w do not satisfy condition
(B) and of course Γ does not satisfy condition (A).
Observe that if w = (w1, . . . , wn) ∈ Zn ⊂ Γn, then Zw1+ · · ·+Zwn ⊂
Z ⊂ R and so Γ and w satisfy condition (B).
D. Holik, M. Karaś 217
Example 2. Let Γ = Z2 with the following operation + : Z2 × Z2 ∋
((a1, a2), (b1, b2)) 7→ (a1+b1, a2+b2) ∈ Z2 and with the order relation de-
fined by (a1, a2) ≤ (b1, b2) if and only if a1 < b1 or (a1 = b1 and a2 ≤ b2).
One can check that Γ is totally ordered additive group. Moreover we
have Γ>0 = {(a1, a2) ∈ Z2 : a1 > 0 or (a1 = 0, a2 > 0)} and so taking
a = (0, 1) and b = (1, 0) we have a, b ∈ (Za + Zb) ∩ Γ>0 and ka < b
for any k ∈ N. Thus, by Lemma 4(2) Γ and w = (a, b) do not satisfy
condition (B) and of course Γ does not satisfy condition (A).
The next example shows the group Γ which is not isomorphic to
(Z,+) but satisfy the conditions (A) and (B).
Example 3. Let Z(2) denotes the localization of Z with respect to the
set {1, 2, 4, . . .} of powers of 2, i.e. Z(2) =
{
k
2l
∈ Q | k ∈ Z, l ∈ N
}
. The
group Γ = Z(2) with natural relation ≤ is of course totally ordered ad-
ditive group. Moreover the set Γ>0 does not have minimal element but
Γ satisfies condition (A). Indeed, if w1, . . . , wn ∈ Γ are any elements,
then taking l = max{l1, . . . , ln}, where wi = ki
2li
, ki ∈ Z, li ∈ N, for
i = 1, . . . , n one can check that Zw1 + · · · + Zwn ⊂ Z 1
2l
and the last
subgroup of Γ is isomorphic to (Z,+) (in the category of ordered group).
In the subsequent examples we show that assumptions of Theorem 1
are essential.
Example 4. Let Γ = Z, w = (2, 2, 1) ∈ Z3 and F1 = x1 + x23, F2 =
x2 + x23. Then dF1 ∧ dF2 = (dx1 + 2x3dx3) ∧ (dx2 + 2x3dx3) = dx1 ∧
dx2 +2x3dx1 ∧ dx3 − 2x3dx2 ∧ dx3. Thus, all assumptions of Theorem 1,
except the inequality (7), are satisfied. Obviously, F1, F2 /∈ k[x1, x2].
Example 5. Let Γ = Z, w = (1, 1, 1) ∈ Z3 and F1 = x1 + x23, F2 =
x2+x23. Then dF1∧dF2 = dx1∧dx2+2x3dx1∧dx3−2x3dx2∧dx3. Thus,
all assumptions of Theorem 1, except that degw(dF1 ∧ dF2) is minimal
possible, are satisfied. Obviously, F1, F2 /∈ k[x1, x2].
Example 6. Let Γ = Z, w = (1, 1, 1, 1) ∈ Z4, F1 = x1 + x2 + x24,
F2 = x1 + x3 + x24 and F3 = F1 + F2. Then, dF1 ∧ dF2 ∧ dF3 = 0. Thus,
all assumptions of Theorem 1, except linear independence of linear parts
of F1, F2, F3, are satisfied. Obviously, F1, F2, F3 /∈ k[x1, x2, x3].
Acknowledgements
This research turned into supported by the AGH University of Krakow
under grant no. 16.16.420.054, funded by the Polish Ministry of Science
and Higher Education.
218 Variable lemma for polynomial rings
References
[1] Edo, E., Kanehira, T., Karaś, M., Kuroda, S.: Separability of wild automorphisms
of a polynomial ring. Transform. Groups 18(1), 81–96 (2013). https://doi.org/10.
1007/s00031-013-9212-2
[2] Hartshorne, R.: Algebraic Geometry. In: Hersh, P., Vakil, R., Wunsch, J. (eds.)
Graduate Texts in Mathematics, vol. 52. Springer New York, NY (1977). https://
doi.org/10.1007/978-1-4757-3849-0
[3] Karaś, M.: There is no tame automorphism of C3 with multidegree (3, 4, 5). Proc.
Amer. Math. Soc. 139(3), 769–775 (2011). https://doi.org/10.1090/S0002-9939-
2010-10779-7
[4] Karaś, M.: Tame automorphisms of C3 with multidegree of the form (3, d2, d3).
J. Pure Appl. Algebra 214(12), 2144–2147 (2010). https://doi.org/10.1016/j.jpaa.
2010.02.017
[5] Karaś, M.: Multidegrees of tame automorphisms of Cn. Diss. Math. 477 (2011)
[6] Kuroda, S.: On the Karaś type theorems for the multidegrees of polynomial auto-
morphisms. J. Algebra 423, 441–465 (2015). https://doi.org/10.1016/j.jalgebra.20
14.10.024
[7] Shafarevich, I.R.: Basic Algebraic Geometry 1. Varieties in Projective Space.
Springer Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37956-7
[8] Shestakov, I.P., Umirbaev, U.U.: The tame and the wild automorphisms of polyno-
mial rings in three variables. J. Amer. Math. Soc. 17, 197–227 (2004). https://doi.
org/10.1090/S0894-0347-03-00440-5
[9] Walters, P.: An Introduction to Ergodic Theory. In: Hersh, P., Wunsch, J., Va-
kil, R. (eds.) Graduate Texts in Mathematics, vol. 79. Springer New York, NY
(1982)
Contact information
D. Holik,
M. Karaś
AGHUniversity of Krakow, Faculty of Applied
Mathematics al. A. Mickiewicza 30, 30-059
Kraków, Poland
E-Mail: holikd@agh.edu.pl,
mkaras@agh.edu.pl
Received by the editors: 22.08.2025
and in final form 20.06.2026.
https://doi.org/10.1007/s00031-013-9212-2
https://doi.org/10.1007/s00031-013-9212-2
https://doi.org/10.1007/978-1-4757-3849-0
https://doi.org/10.1007/978-1-4757-3849-0
https://doi.org/10.1090/S0002-9939-2010-10779-7
https://doi.org/10.1090/S0002-9939-2010-10779-7
https://doi.org/10.1016/j.jpaa.2010.02.017
https://doi.org/10.1016/j.jpaa.2010.02.017
https://doi.org/10.1016/j.jalgebra.2014.10.024
https://doi.org/10.1016/j.jalgebra.2014.10.024
https://doi.org/10.1007/978-3-642-37956-7
https://doi.org/10.1090/S0894-0347-03-00440-5
https://doi.org/10.1090/S0894-0347-03-00440-5
Daria Holik* and Marek Karaś
|
| id | admjournalluguniveduua-article-2428 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-09T01:00:10Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| resource_txt_mv | admjournalluguniveduua/a6/81aa2e9e3ad9d2a263457437eced78a6.pdf |
| spelling | admjournalluguniveduua-article-24282026-07-08T07:55:33Z Variable lemma for polynomial rings Holik, Daria Karaś, Marek polynomial automorphism, tame (wild) automorphism, (weighted) multidegree, general algebraic group Primary: 13N05, 14R10. Secondary: 13B25, 13N15, 14L40 Let \(k\) be a field of characteristic zero, and let \(k[x_1,\ldots,x_n]\) be a polynomial ring in \(n\) variables, where \(n\geq 3\) is an arbitrary positive integer. Assume that we have \(l\in\mathbb{N}\) algebraically independent polynomials \(F_1,\ldots,F_l\in k[x_1,\ldots,x_n]\) with \(2\leq l &lt;n.\) In this paper, we prove that if linear parts of polynomials \(F_1,\ldots,F_l\) are linearly independent and depend only on variables \(x_1,\ldots,x_l\) and the polynomials \(F_1,\ldots,F_l\) meet some weighted-differential criteria then actually \(F_1,\ldots,F_l\in k[x_1,\ldots,x_l].\) Lugansk National Taras Shevchenko University AGH University of Krakow under grant no. 16.16.420.054, funded by the Polish Ministry of Science and Higher Education 2026-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2428 10.12958/adm2428 Algebra and Discrete Mathematics; Vol 41, No 2 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2428/pdf Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | polynomial automorphism tame (wild) automorphism (weighted) multidegree general algebraic group Primary: 13N05 14R10. Secondary: 13B25 13N15 14L40 Holik, Daria Karaś, Marek Variable lemma for polynomial rings |
| title | Variable lemma for polynomial rings |
| title_full | Variable lemma for polynomial rings |
| title_fullStr | Variable lemma for polynomial rings |
| title_full_unstemmed | Variable lemma for polynomial rings |
| title_short | Variable lemma for polynomial rings |
| title_sort | variable lemma for polynomial rings |
| topic | polynomial automorphism tame (wild) automorphism (weighted) multidegree general algebraic group Primary: 13N05 14R10. Secondary: 13B25 13N15 14L40 |
| topic_facet | polynomial automorphism tame (wild) automorphism (weighted) multidegree general algebraic group Primary: 13N05 14R10. Secondary: 13B25 13N15 14L40 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2428 |
| work_keys_str_mv | AT holikdaria variablelemmaforpolynomialrings AT karasmarek variablelemmaforpolynomialrings |