Closed polynomials and saturated subalgebras of polynomial algebras
The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomia...
Gespeichert in:
| Datum: | 2007 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3414 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509501704110080 |
|---|---|
| author | Arzhantsev, I. V. Petravchuk, A. P. Аржанцев, І.В. Петравчук, А. П. |
| author_facet | Arzhantsev, I. V. Petravchuk, A. P. Аржанцев, І.В. Петравчук, А. П. |
| author_sort | Arzhantsev, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:47Z |
| description | The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$. |
| first_indexed | 2026-03-24T02:42:07Z |
| format | Article |
| fulltext |
UDC 512.745
I. V. Arzhantsev∗ (Moscow State Univ., Russia),
A. P. Petravchuk (Kyiv Taras Shevchenko Univ., Ukraine)
CLOSED POLYNOMIALS AND SATURATED SUBALGEBRAS
OF POLYNOMIAL ALGEBRAS
ЗАМКНЕНI ПОЛIНОМИ ТА НАСИЧЕНI ПIДАЛГЕБРИ
ПОЛIНОМIАЛЬНИХ АЛГЕБР
The behavior of closed polynomials, i.e., polynomials f ∈ k[x1, . . . , xn] \ k such that the subalgebra
k[f ] is integrally closed in k[x1, . . . , xn], is studied under extensions of the ground field. Using some
properties of closed polynomials, we prove that every polynomial f ∈ k[x1, . . . , xn] \ k after shifting by
constants can be factorized in a product of irreducible polynomials of the same degree. Some types of
saturated subalgebras A ⊂ k[x1, . . . , xn] are considered, i.e., such that for any f ∈ A \ k a generative
polynomial of f is contained in A.
Дослiджено поведiнку замкнених полiномiв, тобто таких полiномiв f ∈ k[x1, . . . , xn] \ k, що
пiдалгебра k[f ] є iнтегрально замкненою в k[x1, . . . , xn], у випадку розширень основного по-
ля. З використанням деяких властивостей замкнених полiномiв доведено, що кожен полiном
f ∈ k[x1, . . . , xn] \k пiсля зсувiв на константи може бути розкладений у добуток незвiдних полiно-
мiв одного й того ж степеня. Розглянуто деякi типи насичених пiдалгебр A ⊂ k[x1, . . . , xn], тобто
таких алгебр, що для будь-якого f ∈ A \ k породжуючий полiном для f мiститься в A.
1. Introduction. Recall that a polynomial f ∈ k[x1, . . . , xn] \ k is called closed if the
subalgebra k[f ] is integrally closed in k[x1, . . . , xn]. It turns out that a polynomial f is
closed if and only if f is non-composite, i.e., f cannot be presented in the form f = F (g)
for some g ∈ k[x1, . . . , xn] and F (t) ∈ k[t], deg(F ) > 1. Because any polynomial in
n variables can be obtained from a closed polynomial by taking a polynomial in one
variable from it, the problem of studying closed polynomials is of interest. Besides,
closed polynomials in two variables appear in a natural way as generators of rings of
constants of non-zero derivations.
Let us go briefly through the content of the paper. In Section 2 we collect numerous
characterizations of closed polynomials (Theorem 1). A major part of these characteri-
zations is contained in the union of [1 – 4], etc, but some results seem to be new. In
particular, implication (i) ⇒ (iv) in Theorem 1 over any perfect field and Proposition 1
solve a problem stated in [1] (Section 8).
Define a generative polynomial h of a polynomial f ∈ k[x1, . . . , xn] \ k as a closed
polynomial such that f = F (h) for some F ∈ k[t]. Clearly, a generative polynomial
exists for any f. Moreover, a generative polynomial is unique up to affine transformations
(Corollary 1).
The above-mentioned results allow us to prove that over an algebraically closed field
k for any f ∈ k[x1, . . . , xn] \ k and for all but finite number µ ∈ k the polynomial
f + µ can be decomposed into a product f + µ = α · f1µ · f2µ . . . fkµ, α ∈ k×, k > 1,
of irreducible polynomials fiµ of the same degree d not depending on µ and such that
∗ Supported by GK 02.445.11.7407 (Russia).
c© I. V. ARZHANTSEV, A. P. PETRAVCHUK, 2007
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12 1587
1588 I. V. ARZHANTSEV, A. P. PETRAVCHUK
fiµ − fjµ ∈ k, i, j = 1, . . . , k (Corollary 2). This result may be considered as an
analogue of the Fundamental Theorem of Algebra for polynomials in many variables.
Moreover, Stein – Lorenzini – Najib’s Inequality (Theorem 2) implies that the number
of “exceptional” values of µ is less then deg(f). The same inequality gives an estimate
of the number of irreducible factors in f + µ for exceptional µ, see Theorem 3.
Section 4 is devoted to saturated subalgebras A ⊂ k[x1, . . . , xn], i.e., such that for
any f ∈ A \ k a generative polynomial of f is contained in A. Clearly, any subalgebra
that is integrally closed in k[x1, . . . , xn] is saturated. On the other hand, it is known
that for monomial subalgebras these two conditions are equivalent. In Theorem 4
we characterize subalgebras of invariants A = k[x1, . . . , xn]G, where G is a finite
group acting linearly on k[x1, . . . , xn], with A being saturated. This result provides
many examples of saturated homogeneous subalgebras that are not integrally closed in
k[x1, . . . , xn].
2. Characterizations of closed polynomials. Let k be an arbitrary field.
Proposition 1. Let f ∈ k[x1, . . . , xn] \ k and k ⊂ L be a separable extension of
fields. Then f is closed over k if and only if f is closed over L.
Proof. If f = F (h) over k, then the same decomposition holds over L.
Now assume that f is closed over k. Consider an element g ∈ L[x1, . . . , xn] integral
over L[f ]. We shall prove that g ∈ L[f ]. Since the number of non-zero coefficients of
g is finite, we may assume that L is a finitely generated extension of k. Then there
exists a finite separable transcendence basis of L over k, i.e., a finite set {ξ1, . . . , ξm}
of elements in L that are algebraically independent over k and L is a finite separable
algebraic extension of L1 = k(ξ1, . . . ξm).
Let us show that f is closed over L1. The subalgebra k[f ][ξ1, . . . , ξm] is integrally
closed in k[x1, . . . , xn][ξ1, . . . , ξm] [5] (Chapter V.1, Proposition 12). Let T be the set
of all non-zero elements of k[ξ1, . . . , ξm]. Then the localization T−1k[f ][ξ1, . . . , ξm] is
integrally closed in T−1k[x1, . . . , xn][ξ1, . . . ξm] [5] (Chapter V.1, Proposition 16). This
proves that L1[f ] is integrally closed in L1[x1, . . . , xn].
Fix a basis {ω1, . . . , ωk} of L over L1. With any element l ∈ L one may associate an
L1-linear operator M(l) : L → L, M(l)(ω) = lω. Let tr(l) be the trace of this operator.
It is known that there exists a basis {ω?
1 , . . . , ω?
k} of L over L1 such that tr(ωiω
?
j ) = δij
[5] (Chapter V.1.6). Assume that g =
∑
i
ωiai with ai ∈ L1[x1, . . . , xn]. Any ω?
j is
integral over L1 and thus over L1[f ]. This shows that gω?
j is integral over L1[f ]. Set
K = L1(x1, . . . , xn). The element gω?
j determines a K-linear map L⊗K K → L⊗K K,
b → gω?
j b. Since gω?
j is integral over L1[f ], the trace of this K-linear operator is also
integral over L1[f ] [5] (Chapter V.1.6). Note that tr(gω?
j ) =
∑
i
ai tr(ωiω
?
j ). On the
other hand, the elements {ω1 ⊗ 1, . . . , ωk ⊗ 1} form a basis of L⊗K K over K. Hence
tr (ωiω
?
j ) = δij and tr (gω?
j ) = aj is integral over L1[f ]. This shows that aj ∈ L1[f ] for
any j and thus g ∈ L[f ].
The proposition is proved.
Let M be the set of all subalgebras k[f ], f ∈ k[x1, . . . , xn] \ k, partially ordered by
inclusion.
In the next Theorem various characterizations of closed polynomials are collected
(see [1 – 4], etc). A new result here is the implication (i) ⇒ (iv).
Theorem 1. The following conditions on a polynomial f ∈ k[x1, . . . , xn] \ k are
equivalent:
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
CLOSED POLYNOMIALS AND SATURATED SUBALGEBRAS OF POLYNOMIAL ALGEBRAS 1589
(i) f is non-composite;
(ii) k[f ] is a maximal element of M;
(iii) f is closed;
(iv) (k is a perfect field) f + λ is irreducible over k for all but finitely many λ ∈ k;
(v) (k is a perfect field) there exists λ ∈ k such that f + λ is irreducible over k;
(vi) (char k = 0) there exists a (finite) family of derivations {Di} of the algebra
k[x1, . . . , xn] such that k[f ] = ∩iKer Di.
Proof. (i) ⇒ (iv). Let us assume that k = k. Consider a morphism φ : kn → k1,
φ(x1, . . . , xn) = f(x1, . . . , xn). We should prove that all fibers of this morphism except
for finitely many are irreducible. But it follows from the first Bertini theorem (see, for
example, [6, p. 139]).
If a perfect field k is non-closed, then Proposition 1 shows that f ∈ k[x1, . . . , xn] is
closed over k implies that f is closed over k.
The theorem is proved.
Example 1 [1]. If the field k is not perfect, then we can not guarantee that a
polynomial f which is closed over k, will be closed over k as well. Indeed, let
F = k(η) with η /∈ k, ηp ∈ k. The polynomial f(x1, x2) = xp
1 + ηpxp
2 is closed over k.
However, one has a decomposition f = (x1 + ηx2)p over F. The same example works
for (i) 6⇒ (iv) in this case.
Now we are going to show that a generative polynomial is unique up to affine
tarnsformations. Here we need two auxiliary lemmas.
Lemma 1. For any f ∈ k[x1, . . . , xn] \ k, the integral closure A of k[f ] in
k[x1, . . . , xn] has the form A = k[h] for some closed h ∈ k[x1, . . . , xn].
Proof. Since tr.degkQ(A) = 1, we have by the theorem of Gordan (see for
example [4, p. 15]) Q(A) = k(h) for some rational function h. The subfield Q(A)
contains non-constant polynomials, so by the theorem of E. Noether (see for example [4,
p. 16]) the generator h of the subfield Q(A) can be chosen as a polynomial. Note that
k(h) ∩ k[x1, . . . , xn] = k[h] because any rational function (but polynomial) of a non-
constant polynomial cannot be a polynomial. Therefore A ⊆ k[h]. Since the element
h is integral over A and A is integrally closed in k[x1, . . . , xn], we have h ∈ A and
A = k[h].
The lemma is proved.
Note that in the case char k = 0 this lemma follows immediately from the result of
Zaks [7].
Lemma 2. Let k be a field. Polynomials f, g ∈ k[x1, . . . , xn]\k are algebraically
dependent (over k) if and only if there exists a closed polynomial h ∈ k[x1, . . . , xn]
such that f, g ∈ k[h].
Proof. Assume that f, g are algebraically dependent. By the Noether Normalization
Lemma, there exists an element r ∈ k[f, g] such that k[r] ⊂ k[f, g] is an integral
extension. By Lemma 1, the integral closure of k[r] in k[x1, . . . , xn] has a form k[h]
for some closed polynomial h.
Conversely, if f, g ∈ k[h] then these polynomials are obviously algebraically
dependent.
The lemma is proved.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
1590 I. V. ARZHANTSEV, A. P. PETRAVCHUK
Corollary 1. Let f ∈ k[x1, . . . , xn] \ k. The integral closure of the subalgebra
k[f ] in k[x1, . . . , xn] coincides with k[h], where h is a generative polynomial of f. In
particular, a generative polynomial of f exists and is unique up to affine transformations.
3. A factorization theorem. Let us assume in this section that the ground field k is
algebraically closed. Theorem 1 states that for a closed polynomial h ∈ k[x1, . . . , xn]
the polynomial h + λ may be reducible only for finitely many λ ∈ k. Denote by E(h)
the set of λ ∈ k such that h + λ is reducible and by e(h) the cardinality of this set.
Stein’s inequality claims that
e(h) < deg h.
Now for any λ ∈ k consider a decomposition
h + λ =
n(λ,h)∏
i=1
h
dλ,i
λ,i
with hλ,i being irreducible. A more precise version of Stein’s inequality is given in the
next theorem.
Theorem 2 [Stein – Lorenzini – Najib’s inequality]. Let h ∈ k[x1, . . . , xn] be a clos-
ed polynomial. Then ∑
λ
(n(λ, h)− 1) < min
λ
(
∑
i
deg(hλ,i)).
This inequality has rather long history. Stein [8] proved his inequality in characteristic
zero for n = 2. For any n over k = C this inequality was proved in [9]. In 1993,
Lorenzini [10] obtained the inequality as in Theorem 2 in any characteristic, but only for
n = 2 (see also [11] and [12]). Finally, in [13] the proof for an arbitrary n was reduced
to the case n = 2.
Now take any f ∈ k[x1, . . . , xn] \ k, µ ∈ k and consider a decomposition
f + µ = α ·
n(µ,f)∏
i=1
f
dµ,i
µ,i
with α ∈ k× and fµ,i being irreducible.
Let us state the main result of this section.
Theorem 3. Let f ∈ k[x1, . . . , xn] \ k. There exists a finite subset E(f) =
= {µ1, . . . , µe(f) | µi ∈ k} with e(f) < deg f such that:
(1) for any µ /∈ E(f) one has f + µ = α · fµ,1 · fµ,2 . . . fµ,k, where all fµ,i are
irreducible and fµ,i − fµ,j ∈ k;
(2) fµ,i − fν,j ∈ k× for any µ, ν /∈ E(f) with ν 6= µ; in particular, the degree
d = deg(fµ,i) does not depend on i and µ;
(3) deg(fµ,i) ≤ d for any µ ∈ k;
(4)
∑
µ
(
n(µ, f)− deg(f)
d
)
< min
µ
(∑n(µ,f)
i=1
deg(fµ,i)
)
.
Proof. Let h be the generative polynomial of f and f = F (h). Then
F (h) + µ = α · (h + λµ,1) . . . (h + λµ,k)
for some λµ,1, . . . , λµ,k ∈ k. Hence for any µ with λµ,1, . . . , λµ,k /∈ E(h) we have a
decomposition of f + µ as in (1). Note that λµ,i 6= λν,j for µ 6= ν. This proves (2) with
d = deg(h) and gives the inequalities
e(f) ≤ e(h) < deg(h) ≤ deg(f).
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
CLOSED POLYNOMIALS AND SATURATED SUBALGEBRAS OF POLYNOMIAL ALGEBRAS 1591
Any fµ,i is a divisor of some h + λ. This implies (3).
Finally, (4) may be obtained as:∑
µ
(
n(µ, f)− deg(f)
d
)
≤
∑
λ
(n(λ, h)− 1) <
< min
λ
(∑
i
deg(hλ,i)
)
≤ min
µ
∑
j
deg(fµ,j)
.
The theorem is proved.
Remark 1. It follows from the proof of Theorem 3 that
E(f) =
{
−F (−λ) | λ ∈ E(h)
}
;
if f is not closed, then e(f) <
1
2
deg(f).
Corollary 2. Let f ∈ k[x1, . . . , xn] \ k. Then for all but finite number µ ∈ k, the
polynomial f + µ can be decomposed into the product
f + µ = α · f1µ · f2µ . . . fkµ, α ∈ k×, k > 1,
of irreducible polynomials fiµ of the same degree d not depending on the number µ and
such that fiµ − fjµ ∈ k, i, j = 1, . . . , k. The number of exceptional µ
′
s for which such
a decomposition does not exist is at most deg f − 1.
Example 2. Take f(x1, x2) = x2
1x
4
2 − 2x2
1x
3
2 + x2
1x
2
2 + 2x1x
3
2 − 2x1x
2
2 + x2
2 + 1.
Here h = x1x2(x2 − 1) + x2 and F (t) = t2 + 1. It is easy to check that E(h) =
= {0,−1}, thus E(f) = {−1,−2}. We have decompositions:
µ = −1: f − 1 = x2
2(x1x2 − x1 + 1)2;
µ = −2: f − 2 = (x2 − 1)(x1x2 + 1)(x1x2(x2 − 1) + x2 + 1);
µ 6= −1,−2: f+µ =
(
x1x2(x2−1)+x2+λ
)(
x1x2(x2−1)+x2−λ
)
, λ2 = −1−µ.
In this case deg(f) = 6, d = 3,
∑
µ
(n(µ, f)− 2) = 1 and
min
µ
(∑
i
deg(fµ,i)
)
= min{3, 6, 6} = 3.
4. Saturated subalgebras and invariants of finite groups. Let k be a field.
Definition 1. A subalgebra A ⊆ k[x1, . . . , xn] is said to be saturated if for any
f ∈ A \ k the generative polynomial of f is contained in A.
Clearly, the intersection of a family of saturated subalgebras in k[x1, . . . , xn] is again
a saturated subalgebra. So we may define the saturation S(A) of a subalgebra A as the
minimal saturated subalgebra containing A.
If A is integrally closed in k[x1, . . . , xn], then A is saturated. By Theorem 1, if
A = k[f ], then the converse is true. Moreover, the converse is true if A is a monomial
subalgebra. In order to prove it, consider a submonoid P (A) in Zn
≥0 consisting of
multidegrees of all monomials in A. Then monomials corresponding to elements of
the “saturated” semigroup P ′(A) = (Q≥0P (A)) ∩ Zn
≥0 are generative elements of A.
On the other hand, it is a basic fact of toric geometry that the monomial subalgebra
corresponding to P ′(A) is integrally closed in k[x1, . . . , xn], see for example [14]
(Section 2.1).
Now we come from monomial to homogeneous saturated subalgebras. The degree of
monomials deg(αxi1
1 . . . xin
n ) = i1 + . . . + in defines a Z≥0-grading on the polynomial
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
1592 I. V. ARZHANTSEV, A. P. PETRAVCHUK
algebra k[x1, . . . , xn]. Recall that a subalgebra A ⊂ k[x1, . . . , xn] is called homogeneous
if for any element a ∈ A all its homogeneous components belong to A.
Consider a subgroup G ⊂ GLn(k). The linear action G : k[x1, . . . , xn] → k[x1, . . .
. . . , xn] determines the homogeneous subalgebra k[x1, . . . , xn]G of G-invariant polynomi-
als.
Theorem 4. Let G ⊆ GLn(k) be a finite subgroup. The subalgebra A =
= k[x1, . . . , xn]G is saturated in k[x1, . . . , xn] if and only if G admits no non-trivial
homomorphisms G → k×.
Proof. Assume that there is a non-trivial homomorphism φ : G → k×. Let Gφ be
the kernel of φ and Gφ = G/Gφ. Then Gφ is a finite cyclic group of some order k and
it may be identified with a subgroup of k×.
Lemma 3. Let H be a cyclic subgroup of order k in k×. Then any finite dimensi-
onal (over k) H-module W is a direct sum of one-dimensional submodules.
Proof. The polynomial Xk − 1 annihilates the linear operator P in GL(W )
corresponding to a generator of H. By assumption, Xk − 1 is a product of k non-
proportional linear factors in k[X]. This shows that the operator P is diagonalizable.
Lemma 4. Let H ⊂ G be a proper subgroup. Then k[x1, . . . , xn]H 6= k[x1, . . .
. . . , xn]G.
Proof. Let K be a field and G a finite group of its automorphisms. By Artin’s
Theorem [15] (Section 2.1, Theorem 1.8), KG ⊂ K is a Galois extension and [K : KG] =
= |G|. This implies k(x1, . . . , xn)H 6= k(x1, . . . , xn)G. The implication
f
h
∈ k(x1, . . . , xn)G =⇒
f
∏
g∈G,g 6=e g · f
h
∏
g∈G,g 6=e g · f
∈ k(x1, . . . , xn)G
shows that k(x1, . . . , xn)G (resp. k(x1, . . . , xn)H ) is the quotient field of k[x1, . . . , xn]G
(resp. k[x1, . . . , xn]H ), thus k[x1, . . . , xn]H 6= k[x1, . . . , xn]G.
The lemma is proved.
Now we may take a finite-dimensional G-submodule W ⊂ k[x1, . . . , xn]Gφ which
is not contained in k[x1, . . . , xn]G. Then W is a Gφ-module. By Lemma 3, one may
find a Gφ-eigenvector h ∈ W, h /∈ k[x1, . . . , xn]G. Then hk ∈ k[x1, . . . , xn]G and
k[x1, . . . , xn]G is not saturated.
Conversely, assume that any homomorphism χ : G → k is trivial. If h is a generative
element of a polynomial f ∈ k[x1, . . . , xn]G, then for any g ∈ G the element g · h is
also a generative element of f. By Corollary 1, the generative element is unique up to
affine transformation. Without loss of generality we can assume that the constant term
of h is zero. Then the element g ·h has obviously zero constant term and by Corollary 1
this element is proportional to h for any g ∈ G. Thus G acts on the line 〈h〉 via some
character. But any character of G is trivial, so h ∈ k[x1, . . . , xn]G, and k[x1, . . . , xn]G
is saturated.
The theorem is proved.
Remark 2. Since all coefficients of the polynomial
Ff (T ) =
∏
g∈G
(T − g · f)
are in k[x1, . . . , xn]G, any element f ∈ k[x1, . . . , xn] is integral over k[x1, . . . , xn]G.
Thus Theorem 4 provides many saturated homogeneous subalgebras that are not integrally
closed in k[x1, . . . , xn].
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
CLOSED POLYNOMIALS AND SATURATED SUBALGEBRAS OF POLYNOMIAL ALGEBRAS 1593
Corollary 3. Assume that k is algebraically closed and char k = 0.
(1) The subalgebra k[x1, . . . , xn]G is saturated in k[x1, . . . , xn] if and only if G
coincides with its commutant.
(2) The saturation of k[x1, . . . , xn]G is k[x1 . . . , xn] if and only if G is solvable.
Example 3. In general, the saturation S(A) is not generated by generative elements
of elements of A. Indeed, take any field k that contains a primitive root of unit of degree
six. Let G = S3 be the permutation group acting naturally on k[x1, x2, x3] and A3 ⊂ S3
be the alternating subgroup. The proof of Theorem 4 shows that any generative element
of an S3-invariant is an S3-semiinvariant and thus belongs to k[x1, x2, x3]A3 . On the
other hand, S(k[x1, x2, x3]S3) = k[x1, x2, x3].
Example 4. It follows from Theorem 4 that the property of a subalgebra to be
saturated is not preserved under field extensions. Let us give an explicit example of this
effect.
Let k = R and G be the cyclic group of order three acting on R2 by rotations. We
begin with calculation of generators of the algebra of invariants R[x, y]G. Consider the
complex polynomial algebra C[x, y] = R[x, y]⊕iR[x, y] with the natural G-action. Then
C[x, y]G = R[x, y]G⊕ iR[x, y]G. Put z = x+ iy, z = x− iy. Clearly, C[x, y] = C[z, z],
and G acts on z, z as z → εz, z → εz, where ε3 = 1. This implies C[z, z]G =
= C[f1, f2, f3] with f1 = z3, f2 = z3 and f3 = zz. Finally, R[x, y]G = R
[
Re(fi),
Im(fi); i = 1, 2, 3
]
= R
[
x3 − 3xy2, y3 − 3x2y, x2 + y2
]
.
By Theorem 4, the subalgebra R[x, y]G is saturated in R[x, y]. On the other hand,
the subalgebra C
[
x3− 3xy2, y3− 3x2y, x2 + y2
]
contains x3− 3xy2 + i(y3− 3x2y) =
= (x− iy)3.
1. Ayad M. Sur les polynômes f(X, Y ) tels que K[f ] est intégralement fermé dans K[X, Y ] // Acta
arithm. – 2002. – 105, № 1. – P. 9 – 28.
2. Nowicki A. On the jacobian equation J(f, g) = 0 for polynomials in k[x, y] // Nagoya Math. J. –
1988. – 109. – P. 151 – 157.
3. Nowicki A., Nagata M. Rings of constants for k-derivations in k[x1, . . . , xn] // J. Math. Kyoto Univ.
– 1988. – 28. – P. 111 – 118.
4. Schinzel A. Polynomials with special regard to reducibility. – Cambridge Univ. Press, 2000.
5. Bourbaki N. Elements of mathematics, commutative algebra. – Berlin: Springer, 1989.
6. Shafarevich I. R. Basic algebraic geometry I. – Berlin: Springer, 1994.
7. Zaks A. Dedekind subrings of k[x1, . . . , xn] are rings of polynomials // Isr. J. Math. – 1971. – 9. –
P. 285 – 289.
8. Stein Y. The total reducibility order of a polynomial in two variables // Ibid. – 1989. – 68. –
P. 109 – 122.
9. Cygan E. Factorization of polynomials // Bull. Polish. Acad. Sci. Math. – 1992. – 40. – P. 45 – 52.
10. Lorenzini D. Reducibility of polynomials in two variables // J. Algebra. – 1993. – 156. – P. 65 – 75.
11. Kaliman S. Two remarks on polynomials in two variables // Pacif. J. Math. – 1992. – 154. –
P. 285 – 295.
12. Vistoli A. The number of reducible hypersurfaces in a pencil // Invent. math. – 1993. – 112. –
P. 247 – 262.
13. Najib S. Une généralisation de l’inégalité de Stein – Lorenzini // J. Algebra. – 2005. – 292. –
P. 566 – 573.
14. Fulton W. Introduction to toric varieties // Ann. Math. Stud. – 1993. – 131.
15. Lang S. Algebra. – Revised Third Edition. – Springer, 2002. – 211.
Received 14.05.07
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 12
|
| id | umjimathkievua-article-3414 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:07Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/83/6808d4d2658cb6011f25dda111e27483.pdf |
| spelling | umjimathkievua-article-34142020-03-18T19:53:47Z Closed polynomials and saturated subalgebras of polynomial algebras Замкнені поліноми та насичені підалгебри полiномiальних алгебр Arzhantsev, I. V. Petravchuk, A. P. Аржанцев, І.В. Петравчук, А. П. The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$. Досліджено поведінку замкнених поліномів, тобто таких поліномів $f ∈ k[x_1,…,x_n]∖k$, що пiдалгебра k[f] є інтегрально замкненою в k[x1,..., xn], у випадку розширень основного поля. З використанням деяких властивостей замкнених поліномів доведено, що кожен поліном $f ∈ k[x_1,…,x_n]∖k$ після зсувів на константи може бути розкладений у добуток незвідних поліномів одного й того ж степеня. Розглянуто деякі типи насичених підалгебр $A ⊂ k[x_1,…,x_n]$, тобто таких алгебр, що для будь-якого $f ∈ A∖k$ породжуючий поліном для $f$ міститься в $A$. Institute of Mathematics, NAS of Ukraine 2007-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3414 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 12 (2007); 1587–1593 Український математичний журнал; Том 59 № 12 (2007); 1587–1593 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3414/3571 https://umj.imath.kiev.ua/index.php/umj/article/view/3414/3572 Copyright (c) 2007 Arzhantsev I. V.; Petravchuk A. P. |
| spellingShingle | Arzhantsev, I. V. Petravchuk, A. P. Аржанцев, І.В. Петравчук, А. П. Closed polynomials and saturated subalgebras of polynomial algebras |
| title | Closed polynomials and saturated subalgebras of polynomial algebras |
| title_alt | Замкнені поліноми та насичені підалгебри полiномiальних алгебр |
| title_full | Closed polynomials and saturated subalgebras of polynomial algebras |
| title_fullStr | Closed polynomials and saturated subalgebras of polynomial algebras |
| title_full_unstemmed | Closed polynomials and saturated subalgebras of polynomial algebras |
| title_short | Closed polynomials and saturated subalgebras of polynomial algebras |
| title_sort | closed polynomials and saturated subalgebras of polynomial algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3414 |
| work_keys_str_mv | AT arzhantseviv closedpolynomialsandsaturatedsubalgebrasofpolynomialalgebras AT petravchukap closedpolynomialsandsaturatedsubalgebrasofpolynomialalgebras AT aržancevív closedpolynomialsandsaturatedsubalgebrasofpolynomialalgebras AT petravčukap closedpolynomialsandsaturatedsubalgebrasofpolynomialalgebras AT arzhantseviv zamknenípolínomitanasičenípídalgebripolinomialʹnihalgebr AT petravchukap zamknenípolínomitanasičenípídalgebripolinomialʹnihalgebr AT aržancevív zamknenípolínomitanasičenípídalgebripolinomialʹnihalgebr AT petravčukap zamknenípolínomitanasičenípídalgebripolinomialʹnihalgebr |