Spheres over finite rings and their polynomial maps
The paper grew out of our attempt to describe all polynomial self-maps of the real andcomplex circle as well.
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Інститут математики НАН України
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552798054252544 |
|---|---|
| author | Golasiński, M. Ruiz, F. G. Golasiński, M. Ruiz, F. G. |
| author_facet | Golasiński, M. Ruiz, F. G. Golasiński, M. Ruiz, F. G. |
| author_institution_txt_mv | [
{
"author": "M. Golasiński",
"institution": "aculty of Mathematics and Computer Science University of Warmia and Mazury"
},
{
"author": "F. G. Ruiz",
"institution": "Departamento de Álgebra, Geometría y Topología Facultad de Ciencias, Universidad de Málaga Campus Universitario de Teatinos"
}
] |
| author_sort | Golasiński, M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | The paper grew out of our attempt to describe all polynomial self-maps of the real andcomplex circle as well. |
| first_indexed | 2026-08-04T01:05:09Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 148–164
Marek Golasiński, Francisco Gómez Ruiz
Spheres over finite rings and their
polynomial maps
The paper [8] grew out of our attempt to describe all polynomial self-maps
of the real and complex circle as well.
Introduction. The definition of the n-sphere Sn with n ≥ 0 over
the reals can be extended to arbitrary commutative and unitary rings R
which leads to the n-sphere
Sn(R) = {(r0, . . . , rn) ∈ Rn+1; r20 + · · ·+ r2n = 1}
over R. If R is finite then it is worthwhile to compute its cardinality
](Sn(R)). More generally, if V (Fq) is an affine variety defined over a
finite field Fq, we can not only consider the number ](V (Fq)), but also
](V (Fqm)) for m ≥ 1. These can be nicely encoded by the Hasse-Weil
zeta function of V : ζ(V ;X) = exp(
∑∞
m=1
](V (Fqm ))
m Xm) ∈ Q[[X]] which
satisfies a number of fundamental properties, known as the Weil conjec-
tures, which are known to be true mainly by the work [6] of Deligne.
Like for S1, the circle S1(R) is equipped in an abelian group structure.
Further, S1(−) is a functor from commutative and unitary rings into
abelian group. In particular, for the field Q of rational numbers, points
of S1(Q) are determined by Pythagorean triples and S1(Q) is dense in
the circle S1. If R is a finite ring then S1(R) is a finite abelian group and
it is a natural problem to determine its structure.
In [9], the author considers the group structure in S1(R), with R
being a commutative and unitary ring, determines this structure in the
case when R is either a finite field or the ring Zm of integers modulo m,
and describes the group structure on conic sections.
In particular, by [9], the group S1(R) is cyclic provided R is a field or
the ring Zpk of integers modulo pk for a prime odd number p. Further, in
c© Marek Golasiński, Francisco Gómez Ruiz, 2013
M. Golasiński, F. G. Ruiz 149
[9, p. 54] the author has stated: The case p = 2 is particularly interesting
(or nasty, depending on your point of view [oder lästig, je nachdem, wie
man es sieht]).
The aim of Section 1 is to simplify proofs of some results from [9],
present their generalizations and state in Theorem 2.5:
If p is a prime and k ≥ 1 then
S1(Zpk) ∼=
Z+
pk−1(p−1), if p ≡ 1 (mod 4);
Z+
pk−1(p+1)
, if p ≡ 3 (mod 4);
Z+
2 , if k = 1;
Z+
2 ⊕ Z+
22 ⊕ Z+
2k−2 , if k ≥ 2.
The paper [8] grew out of our attempt to describe all polynomial self-
maps of the real and complex circle as well. Then, some results from
[11, 14, 15] on spheres and their polynomial maps into spheres over any
field has been transfered. In virtue of Wood [14] (see also [5, Chapter
13]) a necessary condition for the existence of a non-constant polynomial
map Sm → Sn of spheres for m ≥ n is that 2k+1 > m ≥ n ≥ 2k for
some k ≥ 0. It was shown in [15] that from the homotopy point of view
nothing is lost by complexifying the problem of which homotopy classes
of maps of spheres contain a polynomial representative. Furthermore in
virtue of [7] any complex polynomial self-map of S2(C) yields a regular
self-map of the sphere S2 in a canonical way. Then Loday [11] using
algebraic and topological K-theory proved some results on polynomials
maps into Sn. For instance, every polynomial map from the torus Tn to
Sn is null-homotopic if n > 1. For n even those results were extended in
[3, 4] to regular and then in [5] to polynomial maps Sn1 ×· · ·×Snk → Sn
with n = n1 + · · ·+ nk odd. Certainly, polynomial maps Sm1(R)× · · · ×
Smk(R) → Tn(R) are worth to be studied from the algebraic point of
view for any field R. We made use of the abelian group structure on
the sphere S1(R) to show in [8, Corollary 2.11] that for any polynomial
self-map f : S1(R) → S1(R) there are α ∈ S1(R) and an integer n such
that f(z) = αzn for any z ∈ S1(R) provided the field R is infinite. All
polynomial maps Sm1(R) × · · · × Smk(R) → Tn(R) are listed in [8] for
any infinite field R.
Section 2 takes up the systematic study of spheres Sn(R) over a fi-
nite field R and polynomial maps Sm1(R) × · · · × Smk(R) → Sn1(R) ×
· · · × Snl(R) with m1, . . . ,mk, n1, . . . , nl ≥ 0. Theorem 3.2 shows the
cardinality ](Sn(R)) of the n-sphere Sn(R):
150 Spheres over finite rings
If the characteristic χ(R) 6= 2 then for any number n ≥ 1 it holds:
]Sn(R) =
{
(]R)n − (]R)
n
2 η((−1)
n
2 ), if n is even;
(]R)n − (]R)
n−1
2 η((−1)
n+1
2 ) if n is odd,
where
η(1) = 1 and η(−1) =
1, if the equation X2 + 1 = 0 has a solution
inR;
−1 otherwise
and Corollary 3.4 asserts that any such any map Sm1(R)×· · ·×Smk(R)→
Sn1(R)× · · · × Snl(R) is a polynomial one.
1. Circles over a finite ring. Let R be a commutative and
unitary ring. The set
S1(R) = {(r0, r1) ∈ R×R; r20 + r21 = 1}
is called the 1-sphere or the circle over R.
Observe that on S1(R) there is an abelian group structure de-
fined by (r0, r1) ◦ (r′0, r
′
1) = (r0r
′
0 − r1r
′
1, r0r
′
1 + r1r
′
0) for any points
(r0, r1), (r′0, r
′
1) ∈ S1(R). Writing SO(2, R) for the group of special or-
thogonal 2× 2-matrices over R, we may easily show
Remark 2.1. (1) For any commutative and unitary ring R there is an
isomorphism of groups
S1(R) ∼= SO(2, R)
determined by the assignment (r0, r1) 7→
(
r0 r1
−r1 r0
)
for (r0, r1) ∈ S1(R).
(2) If R1, R2 are commutative and unitary rings then there is an iso-
morphism of groups S1(R1 ×R2) ∼= S1(R1)× S1(R2).
Next, consider the quotient ringR[i] = R[X]/(X2+1), where i denotes
the class of X in R[X]/(X2 + 1) and write U(R) for the multiplicative
group of R. Let χ(R) denote the characteristic of R. Then, we may state:
Proposition 2.2. F̨or any unitary ring R there is a group monomor-
phism S1(R)→ U(R[i]). Further:
M. Golasiński, F. G. Ruiz 151
(1) if χ(R) = 2 then S1(R) = {(1 + r + s, r); r, s ∈ Rwith s2 = 0}
and there is a splitting short exact sequence
0→ R+ → S1(R)→ R̃→ 1,
where R+ is the additive group of R and the group R̃ = {s ∈ R; s2 = 0}
with s1 ◦ s2 = s1 + s2 + s1s2 for s1, s2 ∈ R̃;
(2) if i ∈ R with i2 = −1 then there is an exact sequence of abelian
groups
0→ R0 → S1(R)→ U(R),
where R0 = {r ∈ R; 2r = 0};
(i) if 2 ∈ U(R) then there a group isomorphism
S1(R)
∼=→ U(R);
(ii) if χ(R) = 2 then there is a splitting short exact sequence
0→ R→ S1(R)→ R1 → 1,
where R1 = {r ∈ R; r2 = 1};
(3) if i 6∈ R then there is an exact sequence
1→ S1(R)→ U(R[i])
ρ→ U(R)
of abelian groups, where ρ(r0 + r1i) = r20 + r21 for r0 + r1i ∈ U(R[i]).
Further, if R is a finite field then U(R[i])
ρ→ U(R) is onto.
Proof. Certainly, the map ϕ : S1(R)→ U(R[i]) given by ϕ(r0, r1) =
r0 + r1i for (r0, r1) ∈ S1(R) is a group monomorphism.
(1) Let χ(R) = 2. If r, s ∈ R with s2 = 0 then (1 + r + s, r) ∈ S1(R).
Conversely, if (r0, r1) ∈ S1(R) then r0 = 1 + r1 + (1 + r0 + r1) and
(1 + r0 + r1)2 = 0. Hence, S1(R) = {(1 + r + s, r); r, s ∈ Rwith s2 = 0}.
Further, one can easily see that the map φ : R+ → S1(R) given by φ(r) =
(1+r, r) for r ∈ R is a group monomorphism. Write R̃ = {s ∈ R; s2 = 0}
and s1 ◦ s2 = s1 + s2 + s1s2 for s1, s2 ∈ R̃. Then, (R̃, ◦) is an abelian
group and the map ρ : S1(R) → R̃ given by ρ(1 + r + s, r) = s for
(1 + r + s, r) ∈ S1(R) is an epimorphism. The sequence
0→ R+ φ→ S1(R)
ρ→ R̃→ 0
is exact and the map ρ′ : R̃→ S1(R) given by ρ′(s) = (1 + s, 0) for s ∈ R̃
determines its splitting.
152 Spheres over finite rings
(2) Write R0 = {r ∈ R; 2r = 0}. Then, the maps
α : R0 → S1(R) and ϕ : S1(R)→ U(R)
given by α(r) = (1+r, r) for r ∈ R0 and ϕ(r0, r1) = r0 +r1i for (r0, r1) ∈
S1(R) are group homomorphisms with Kerα = {0} and Imα = Kerϕ.
Notice that r ∈ U(R) with r+ r−1 = 2s for some s ∈ R implies (s,−(r−
s)i) ∈ S1(R) and ϕ(s,−(r − s)i) = r. Consequently,
Imϕ = {r ∈ U(R); r + r−1 ∈ 2R}.
(i) If 2 ∈ U(R) then R0 = {0} and r + r−1 ∈ Imϕ for r ∈ U(R).
Hence, the map
ψ : U(R)→ S1(R)
given by ψ(r) = (2−1(r−1 + r), 2−1(r−1− r)i) for r ∈ U(R) is the inverse
of the ϕ : S1(R)→ U(R) above.
(ii) If χ(R) = 2 then R0 = R, Imϕ = {r ∈ R; r2 = 1} = R1 and the
short exact sequence
0→ R+ → S1(R)→ R1 → 1
splits as an exact sequence of elementary 2-groups.
(3) Consider the group homomorphism ρ : U(R[i]) → U(R) given by
ρ(r0 + r1i) = r20 + r21 for r0 + r1i ∈ U(R[i]). Then, Kerρ = S1(R) and
consequently we get the required short exact sequence 1 → S1(R) →
U(R[i])→ U(R).
Let now R be a finite field and define the group endomorphism π :
U(R) → U(R) given by π(r) = r2 for r ∈ U(R). If χ(R) = 2 then π is
an automorphism and so U(R[i])
ρ→ U(R) is onto.
Now, suppose that χ(R) 6= 2 and write ]X for the cardinality of a
finite set X. Notice that the group endomorphism U(R) → U(R) given
by r 7→ r2 for r ∈ U(R) leads to kerπ ∼= Z2 and ]{r2; r ∈ U(R)} =
]U(R)
2 . Given r ∈ U(R), we follow [10, Remark 6.25] to consider the sets
A = {r20; r0 ∈ U(R) ∪ {0}} and B = {r − r21; r1 ∈ U(R) ∪ {0}}. Then,
]A = ]B = ]U(R)
2 + 1 and consequently A ∩ B 6= ∅ which implies that
ρ(r0 + r1i) = r.
�
Writing Z+
m for the cyclic group with order m, we deduce (see [9,
Korollar 6]):
M. Golasiński, F. G. Ruiz 153
Corollary 2.3. Įf R is a finite field then there is an isomorphism of
groups:
(1) S1(R) ' (Z+
2 )k provided ]R = 2k and χ(R) = 2;
(2) S1(R) '
{
Z+
]R−1, if ]R ≡ 1 (mod 4);
Z+
]R+1, if ]R ≡ 3 (mod 4).
provided χ(R) 6= 2.
Proof. (1) follows directly from Proposition 2.2(2)(ii).
(2) If ]R ≡ 1 (mod 4) then i ∈ R and by Proposition 2.2(2), we get
an isomorphism S1(R) ∼= U(R). Hence, the well-known isomorphism
U(R) ∼= Z+
]R−1 yields S1(R) ∼= Z+
]R−1.
If ]R ≡ 3 ( mod 4) then, by Fermat Theorem on Sums of Two Squares,
i 6∈ R. Then, by Proposition 2.2(3), there is an exact sequence 1 →
S1(R) → U(R[i]) → U(R) → 1 of abelian groups. Because R and R[i]
are finite fields, there are isomorphisms U(R) ∼= Z]R−1 and U(R[i]) ∼=
Z(]R)2−1. Consequently, we deduce S1(R) ∼= Z+
]R+1 and the proof is
complete.
�
Let now R = Zm, the ring of integers modulo m. The primary
factorization m = pk11 · · · p
kt
t yields an isomorphism of rings Zm
∼=→
Z
p
k1
1
× · · · × Z
p
kt
t
. Because S1(−) is a product preserving functor from
unitary rings to abelian groups, we get an isomorphism
S1(Zm)
∼=→ S1(Z
p
k1
1
)× · · · × S1(Z
p
kt
t
)
and ]S1(Zm) = ]S1(Z
p
k1
1
) · · · ]S1(Z
p
kt
t
). Hence, the problem of determin-
ing the structure of S1(Zm) and ]S1(Zn) has been reduced to the case of
prime powers pk. By the claim in [9, p. 54], the group S1(Zpk) is cyclic
provided p is an odd prime. A proof of that is presented below.
Lemma 2.4. Įf p is a prime and k ≥ 1 then
U(Zpk [i]) ∼=
Z+
pk−1(p−1) ⊕ Z+
pk−1(p−1), if p ≡ 1 (mod 4);
Z+
pk−1 ⊕ Z+
pk−1(p2−1), if p ≡ 3 (mod 4);
Z+
2 , if p = 2 and k = 1;
Z+
22 ⊕ Z+
2k−2 ⊕ Z+
2k−1 , if p = 2 and k ≥ 2.
Proof. First, let p be an odd prime. Recall the well-known the
isomorphism U(Zpk) ∼= ((p) + 1) ⊕ U(Zp) ∼= Z+
pk−1(p−1) stated in [13,
154 Spheres over finite rings
Theorem 6.7], where (p) is the nilpotent principal ideal of Zpk generated
by p.
Let p ≡ 1 (mod 4) and i ∈ U(Zpk) with order four. Because i ∈ Zp−1
and −1 is the only element in Zp−1 with order two, we deduce that
i2 = −1. Consequently, Zpk [i] ∼= Zpk × Zpk and U(Zpk [i]) ∼= U(Zpk) ×
U(Zpk) ∼= Z+
pk(p−1) ⊕ Z+
pk(p−1).
If p ≡ 3 ( mod 4) then, by Fermat’s Theorem on Sums of Two Squares,
i 6∈ Zpk . Given r0 + r1i ∈ Zpk [i], we see that r0 + r1i ∈ U(Zpk [i]) if and
only if r20 +r21 ∈ U(Zpk) or equivalently, if and only if r0 ∈ U(Zpk) or r1 ∈
U(Zpk). Hence, Zpk [i] is a p-primary ring with the nilpotent principal
prime ideal (p) and ](p) = p2(k−1). Then, the residue filed Zpk [i]/(p) ∼=
Zp2 and in view of [2, Proposition 1], we deduce that U(Zpk [i]) ∼= ((p) +
1) ⊕ U(Zp2). Following the proof of [13, Theorem 6.7], we get (1 +
p)p
l−2
, (1+pi)p
l−2 6≡ 1 ( mod pl) and (1+p)p
l−1
, (1+pi)p
l−1 ≡ 1 ( mod pl)
for l ≥ 2. Because
〈
1+p
〉
∩
〈
1+pi
〉
= {1}, we deduce a group isomorphism
((p)+1) ∼=
〈
1+p
〉
⊕
〈
1+pi
〉 ∼= Z+
pk−1 ⊕Z+
pk−1 . Consequently, we get that
U(Zpk [i]) ∼= Z+
pk−1 ⊕ Z+
pk−1(p2−1).
Let now p = 2. First, it is obvious that U(Z2[i]) = {1, i} ∼= Z2.
Hence, we can assume that k ≥ 2. Recall form [13, Theorem 5.44] that
U(Z2k) ∼=
〈
5
〉
⊕
〈
− 1
〉 ∼= Z+
2k−2 ⊕ Z+
2 for k ≥ 2. Because r0 + r1i ∈
U(Z2k [i]) if any only if r0 is odd and r1 is even or vise versa, we get
]U(Z2k [i]) = 22k−1. Further, (1 + 2i)2
l−2 ≡ 2l−1 + 1 + 2l−1i (mod 2l)
for l > 2. This implies that 2k−1 is the order of 1 + 2i. Next, the
intersection of any two of the subgroups
〈
i
〉
,
〈
5
〉
and
〈
1 + 2i
〉
is the
trivial group and ]U(Z2k [i]) = 22k−1. Thus, we deduce that U(Z2k [i]) ∼=〈
i
〉
⊕
〈
5
〉
⊕
〈
1 + 2i
〉 ∼= Z22 ⊕ Z2k−2 ⊕ Z2k−1 for k ≥ 2 and the proof is
complete.
�
Now, we are in a position to show the main result of this Section:
Theorem 2.5. Įf p is a prime and k ≥ 1 then
S1(Zpk) ∼=
Z+
pk−1(p−1), if p ≡ 1 (mod 4);
Z+
pk−1(p+1)
, if p ≡ 3 (mod 4);
Z+
2 , if k = 1;
Z+
2 ⊕ Z+
22 ⊕ Z+
2k−2 , if k ≥ 2.
M. Golasiński, F. G. Ruiz 155
Proof. (1) If p ≡ 1 (mod 4) then i ∈ Zpk . Because 2 ∈ U(Zpk), by
Proposition 2.2(2), the map ρ : S1(Zpk) → U(Zpk) given by ρ(r0, r1) =
r0 + r1i for (r0, r1) ∈ S1(Zpk) is an isomorphism of groups. Thus,
S1(Zpk) ∼= U(Zpk) ∼= Z+
pk−1(p−1).
(2) If p ≡ 3 (mod 4) then i 6∈ Zpk . Further, U(Zpk) ∼= Z+
pk−1(p−1)
and, in view of Lemma 2.4, it holds U(Zpk [i]) ∼= Z+
pk−1 ⊕ Z+
pk−1(p2−1).
Next, consider the map ρ : U(Zpk [i]) → U(Zpk) defined in Propo-
sition 2.2(2). Then, the restriction ρ|Z+
pk−1
is an isomorphism and,
in view of Proposition 2.2(3), the restriction ρ|Z+
p2−1
is onto. Conse-
quently, ρ : U(Zpk [i]) → U(Zpk) is onto and the short exact sequence
1 → S1(Zpk) → U(Zpk [i])
ρ→ U(Zpk) → 1 from Proposition 2.2(3) yields
S1(Zpk) ∼= Z+
pk−1(p+1)
.
(3) For the group homomorphism ρ : U(Z2k [i]) → U(Z2k) given by
ρ(r0 + r1i) = r20 + r21 for r0 + r1i ∈ U(Z2k [i]), by Proposition 2.2(3), we
get the short exact sequence
1→ S1(Z2k)→ U(Z2k [i])
ρ→ U(Z2k)
of abelian groups with k ≥ 1.
Because U(Z2) = {1}, Lemma 2.4 yields that S1(Z2) ∼= U(Z2[i]) ∼=
Z+
2 . If k ≥ 2 then by the proof of Lemma 2.4, we have that U(Z2k [i]) ∼=〈
i
〉
⊕
〈
5
〉
⊕
〈
1 + 2i
〉 ∼= Z22 ⊕ Z2k−2 ⊕ Z2k−1 . Because ρ(i) = 1, ρ(5) = 52,
ρ(1 + 2i) = 5 and U(Z2k) ∼=
〈
5
〉
⊕
〈
− 1
〉 ∼= Z+
2k−2 ⊕ Z+
2 , we deduce that
Im ρ =
〈
5
〉 ∼= Z+
2k−2 . Consequently, the exact sequence
1→ S1(Z2k)→ U(Z2k [i])
ρ→ Z2k−2 → 1
yields S1(Z2k) ∼= Z+
2 ⊕ Z+
22 ⊕ Z+
2k−2 for k ≥ 2 and the proof is complete.
�
2. Spheres over finite fields and their polynomial
maps. Let R be a commutative and unitary ring. Then, we notice:
156 Spheres over finite rings
Remark 3.1. For any commutative and unitary ring R there is a bijec-
tion S3(R) ∼= SU(R[i]) determined by the assignment
(r0, r1, r2, r3) 7→
(
r0 + r1i r2 + r3i
−r2 + r3i r0 − r1i
)
for (r0, r1, r2, r3) ∈ S3(R). Consequently, S3(R) inherits the group struc-
ture from SU(R[i]). Notice that S2(R) ∼= {A ∈ SU(R[i]); tr (A) = 0}
provided 2R = 0, where tr : SU(R[i])→ R[i] is the trace function.
Notice that there is an embedding Rn0 ↪→ Sn(R) given by
(r0, . . . , rn−1) 7→ (1 + r0 + · · ·+ rn−1, r0, . . . , rn−1)
for (r0, . . . , rn−1) ∈ Rn0 , where R0 = {r ∈ R; 2r = 0}. In particular,
Rn ↪→ Sn(R) provided χ(R) = 2. If R is a field with χ(R) = 2 then
certainly there is a bijection Sn(R) ∼= Rn and ]Sn(R) = (]R)n.
Now, suppose that R is a finite field with χ(R) 6= 2. Basing on [10,
Theorems 6.26 and 6.27], we obtain:
Theorem 3.2. Įf R is a finite field with χ(R) 6= 2 then for any number
n ≥ 1 it holds:
]Sn(R) =
{
(]R)n + (]R)
n
2 η((−1)
n
2 ), if n is even;
(]R)n − (]R)
n−1
2 η((−1)
n+1
2 ), if n is odd,
where η(1) = 1 and η(−1) =
1, if the equation x2 + 1 = 0
has a solution in R;
−1, otherwise.
Let ]R = pk for an odd prime p. Notice that η(−1) = 1 if and only if
p ≡ 1 (mod 4) or k is an even number.
To examine polynomial maps P = (P0, . . . , Pn) : Sm(R) → Sn(R) in
that case a general result would be useful.
Proposition 3.3. L̨et R be a field and S ⊆ Rm+1, T ⊆ Rn+1 finite
subsets. Then any map f : S → T is a polynomial one for m,n ≥ 0.
Proof. Given a finite subset S ⊆ Rm+1 there is obviously a finite
subset S0 = {r1, . . . , rk} ⊆ R with S ⊆ Sm+1
0 . It is well-know that there
are interpolation polynomials Pr1(X), . . . , Prk(X) ∈ R[X] with Pri(xj) =
M. Golasiński, F. G. Ruiz 157
δrirj for i, j = 0, . . . , k. Next for any s = (ri0 , . . . , rim) ∈ Sm+1
0 consider
the polynomial
Ps(X0, . . . , Xm) = Pri0 (X0) · · ·Prim (Xm) ∈ R[X0, . . . , Xm].
Then Ps(s′) = δss′ for any s, s′ ∈ Sm+1
0 .
Now, given a map f : S → T write f(s) = (f0(s), . . . , fn(s)) for
any point s ∈ S. Then, the polynomial map S → T determined by
polynomials:
Q0(X0, . . . , Xm) =
∑
s∈S
f0(s)Ps(X0, . . . , Xm),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Qn(X0, . . . , Xm) =
∑
s∈S
fn(s)Ps(X0, . . . , Xm)
coincides with f : S → T and the proof is complete. �
In particular, the following conclusion follows.
Corollary 3.4. L̨et R be a finite field. Then any map Sm1(R) ×
· · · × Smk(R) → Sn1(R) × · · · × Snl(R) is a polynomial one for
m1, . . . ,mk, n1, . . . , nl ≥ 0.
Let EndR(R[X1, . . . , Xn]) be the set of all R-homomorphisms
of R[X1, . . . , Xn] and AutR(R[X1, . . . , Xn]) the group of all its R-
automorphisms. Write T (R,n) for the tame polynomial auto-
morphism subgroup of AutR(R[X1, . . . , Xn]) generated by (X1 +
F (X2, . . . , Xn), X2, . . . , Xn) for all F (X2, . . . , Xn) ∈ R[X2, . . . , Xn],
P(Rn) for the set of all self-maps of Rn and B(Rn) the group of all
bijections of Rn. Then, we get an obvious map
E : EndR(R[X1, . . . , Xn]) −→ P(Rn).
Theorem 3.5. ([12]) L̨et R be a finite field and Fp the simple field,
where p is a prime. Then:
(1) ]E(T (R, 1)) = ]B(R)/]R − 2)!, so E(T (R, 1)) = B(R) only if R =
F2, F3;
(2) if n ≥ 2 and χ(R) 6= 2 or R = F2 then E(T (R,n)) = B(Rn);
(3) if n ≥ 2, χ(R) = 2 and ]R > 2 then ]E(T (R,n) = ]B(Rn)/2. In
fact,
E(T (R,n)) is the alternating subgroup A(Rn) of the group B(Rn).
158 Spheres over finite rings
Now, any bijection of Sn1(R) × · · · × Snl(R) yields an bijection of
Rm1+···+mk+k. Furthermore, for χ(R) = 2 there is an obvious polynomial
isomorphism Sn(R)→ Rn. Consequently, Theorem 3.5 leads to:
Corollary 3.6. L̨et R be a finite field. Then:
(1) if χ(R) 6= 2 or R = F2 then any bijection of B(Sn1(R) × · · · ×
Snl(R)) is an invertible polynomial map;
(2) if ]R > 2 and χ(R) = 2 then any bijection of A(Sn1(R) × · · · ×
Snl(R)) is an invertible polynomial map.
Let R be a commutative and unitary ring. Then, we could consider
the non-commutative and unitary ring R{i, j, k} with i2 = j2 = k2 =
−1, ij = k, jk = i, ki = j. Given q = r0 + r1i + r2j + r3k ∈ R{i, j, k},
we write |q|2 = r20 + r21 + r22 + r23 and q̄ = r0 − r1i − r2j − r3k. Then,
qq̄ = |q|2, |q1q2|2 = |q1|2|q2|2 for q, q1, q2 ∈ R{i, j, k} and
S3(R) ∼= {q ∈ R{i, j, k}; |q|2 = 1}.
Hence, S3(R) inherits the group structure which coincides with the pre-
vious one. Further, we have a group monomorphism
ϕ : S3(R)→ U(R{i, j, k})
given by ϕ(r0, r1, r2, r3) = r0 + r1i+ r2j + r3k for (r0, r1, r2, r3) ∈ S3(R).
Notice that r0 + r1i + r2j + r3k ∈ U(R{i, j, k}) if and only if r20 + r21 +
r22 + r23 ∈ U(R). Hence, the map
ρ : U(R{i, j, k})→ U(R)
given by ρ(r0+r1i+r2j+r3k) = r20 +r21 +r22 +r23 for r0+r1i+r2j+r3k ∈
U(R{i, j, k}) is a well-defined group homomorphism and the sequence
1→ S3(R)
ϕ→ U(R{i, j, k}) ρ→ U(R)
is exact.
Next, we consider the non-associative and unitary ring
R{e1, e2, e3, e4, e5, e6, e7}, where products eset are defined by
the Cayley algebra rules for s, t = 1, 2, 3, 4, 5, 6, 7. Given c =
r0+r1e1+r2e2+r3e3+r4e4+r5e5+r6e6+r7e7 ∈ R{e1, e2, e3, e4, e5, e6, e7},
write |c|2 = r20 + r21 + r22 + r23 + r24 + r25 + r26 + r27. Then, |c1c2|2 = |c1|2|c2|2
for c1, c2 ∈ R{e1, e2, e3, e4, e5, e6, e7} and
S7(R) ∼= {c ∈ R{e1, e2, e3, e4, e5, e6, e7}; |c|2 = 1}
M. Golasiński, F. G. Ruiz 159
inherits a non-associative group structure.
Notice that we have a non-associative group monomorphism
ϕ : S7(R)→ U(R{e1, e2, e3, e4, e5, e6, e7})
given by ϕ(r0, r1, r2, r3, r4, r5, r6, r7) = r0 + r1e1 + r2e2 + r3e3 + r4e4 +
r5e5+r6e6+r7e7 for (r0, r1, r2, r3, r4, r5, r6, r7) ∈ S7(R). Notice that r0+
r1e1+r2e2+r3e3+r4e4+r5e5+r6e6+r7e7 ∈ U(R{e1, e2, e3, e4, e5, e6, e7})
if and only if r20 + r21 + r22 + r23 + r24 + r25 + r26 + r27 ∈ U(R). Hence, the
map
ρ : U(R{e1, e2, e3, e4, e5, e6, e7})→ U(R)
given by ρ(r0 + r1e1 + r2e2 + r3e3 + r4e4 + r5e5 + r6e6 + r7e7) = r20 +
r21 + r22 + r23 + r24 + r25 + r26 + r27 for r0 + r1e1 + r2e2 + r3e3 + r4e4 +
r5e5 + r6e6 + r7e7 ∈ U(R{e1, e2, e3, e4, e5, e6, e7}) is a well-defined non-
associative group homomorphism and the sequence
1→ S7(R)
ϕ→ U(R{e1, e2, e3, e4, e5, e6, e7})
ρ→ U(R)
is exact.
If R1, R2 are commutative and unitary rings then there is a bijection
Sn(R1×R2) ∼= Sn(R1)×Sn(R2) for n ≥ 0. Because the primary factoriza-
tionm = pk11 · · · p
kt
t yields an isomorphism of rings Zm
∼=→ Z
p
k1
1
×· · ·×Z
p
kt
t
,
we derive a bijection
Sn(Zm) ∼= Sn(Z
p
k1
1
)× · · · × Sn(Z
p
kt
t
).
Thus, the study of Sn(Zm) reduces to Sn(Zpk) for any prime p and k ≥ 1.
Proposition 3.7. Įf p is a prime and k ≥ 1 then:
(1) ]S3(Zpk) =
{
p3k−2(p2 − 1), if p is an odd prime;
23k, if p = 2;
(2) ]S7(Zpk) =
{
p7k−4(p2 − 1)(p2 + 1), if p is an odd prime;
27k, if p = 2.
Proof. (1) First, notice that r0 + r1i+ r2j + r3k 6∈ U(Zpk{i, j, k}) if
only if r20 + r21 + r22 + r23 ≡ 0 (mod p) or equivalently, r20 + r21 + r22 + r23 = 0
in the field Zp.
If p is an odd prime then, in view of [10, Theorem 6.26], the equation
r20 + r21 + r22 + r23 = 0 has p3 + (p − 1)p solutions in Zp. Consequently,
160 Spheres over finite rings
the equation r20 + r21 + r22 + r23 ≡ 0 (mod p) has p4(k−1)(p3 + (p − 1)p) =
p4k−3(p2 + p − 1) solutions in Zpk . This implies that ]U(Zpk{i, j, k}) =
p4k − p4k−3(p2 + p− 1) = p4k−3(p2 − 1)(p− 1).
If p = 2 then the equation r20 + r21 + r22 + r23 = 0 has 23 solutions
in Z2. Consequently, the equation r20 + r21 + r22 + r23 ≡ 0 (mod 2) has
24(k−1)23 = 24k−1 solutions in Z2k . This implies that ]U(Z2k{i, j, k}) =
24k − 24k−1 = 24k−1.
Next, by Lagrange Four-Square Theorem, the map ρ :
U(Zpk{i, j, k}) → U(Zpk) is onto for any prime p and k ≥ 1. Hence,
the short exact sequence
1→ S3(Zpk)
ϕ→ U(Zpk{i, j, k})
ρ→ U(Zpk)→ 1
and U(Zpk) ∼=
Zpk−1(p−1), if p is an odd prime;
{1}, if p = 2 and k = 1;
Z2 ⊕ Z2k−2 , if p = 2 and k ≥ 2
lead to (1).
(2) If p is an odd prime then, in view of [10, Theorem 6.26], the
equation r20 + r21 + r22 + r23 + r24 + r25 + r26 + r27 = 0 has p7 + (p − 1)p3
solutions in Zp. Consequently, the equation r20 + r21 + r22 + r23 + r24 +
r25 + r26 + r27 ≡ 0 (mod p) has p8(k−1)(p7 + (p− 1)p3) = p8k−5(p4 + p− 1)
solutions in Zpk . This implies that ]U(Zpk{e1, e2, e3, e4, e5, e6, e7}) =
p8k − p8k−5(p4 + p− 1) = p8k−5(p2 − 1)(p− 1)(p2 + 1).
If p = 2 then the equation r20 + r21 + r22 + r23 + r24 + r25 + r26 + r27 = 0 has
27 solutions in Z2. Consequently, the equation r20 + r21 + r22 + r23 + r24 +
r25 + r26 + r27 ≡ 0 (mod 2) has 28(k−1)27 = 28k−1 solutions in Z2k . This
implies that ]U(Z2k{e1, e2, e3, e4, e5, e6, e7}) = 28k − 28k−1 = 28k−1.
Then, we follow mutatis mutandis the procedure presented in (1) and
the proof is completed. �
Now, for z = r0 + r1i ∈ R[i], we write |z|2 = r20 + r21 and z̄ = r0 − r1i.
Then, zz̄ = |z|2, z ∈ U(R[i]) if and only if |z|2 ∈ U(R) and
S3(R) ∼= {(z0, z1) ∈ R[i]×R[i]; |z0|2 + |z1|2 = 1}.
Notice that there is an action
◦ : S1(R)× S3(R) −→ S3(R)
such that λ ◦ (z0, z1) = (λz0, λz1) for λ ∈ S1(R) and (z0, z1) ∈ S3(R).
M. Golasiński, F. G. Ruiz 161
Next, q ∈ U(R{i, j, k}) if and only if |q|2 ∈ U(R) for q ∈ R{i, j, k},
and
S7(R) ∼= {(q0, q1) ∈ R{i, j, k} ×R{i, j, k}; |q0|2 + |q1|2 = 1}.
Further, there is an action
◦ : S3(R)× S7(R) −→ S7(R)
such that λ ◦ (q0, q1) = (λq0, λq1) for λ ∈ S3(R) and (q0, q1) ∈ S7(R).
Now, we mimic the Hopf maps h : S3 −→ S2 and H : S7 −→ S4 to
define
h(R) : S3(R) −→ S2(R)
by h(R)(z0, z1) = (|z0|2 − |z1|2, 2z0z̄1) for (z0, z1) ∈ S3(R) and
H(R) : S7(R) −→ S4(R)
by H(R)(q0, q1) = (|q0|2 − |q1|2, 2q0q̄1) for (q0, q1) ∈ S7(R).
Proposition 3.8. L̨et R be a local commutative and unitary ring such
that 2 is not a zero divisor of R. Then:
(1) h(R)−1(h(R)(z0, z1)) = {(λz0, λz1); for λ ∈ S1(R)} ∼= S1(R)
for any (z0, z1) ∈ S3(R);
(2) H(R)−1(h(R)(q0, q1)) = {(λq0, λq1); for λ ∈ S3(R)} ∼= S3(R)
for any (q0, q1) ∈ S7(R).
Proof. (1) Let (z0, z1) ∈ S3(R). Then, certainly it holds
{(λz0, λz1); forλ ∈ S1(R)} ⊆ h(R)−1(h(R)(z0, z1)).
Suppose that h(R)(w0, w1) = h(R)(z0, z1) for some (w0, w1) ∈ S3.
Then, |w0|2 − |w1|2 = |z0|2 − |z1|2 and 2w0w̄1 = 2z0z̄1. Because |w0|2 +
|w1|2 = 1 = |z0|2 + |z1|2 and 2 ∈ R is not a zero divisor, we get |w0|2 =
|z0|2, |w1|2 = |z1|2 and w0w̄1 = z0z̄1. Further, R is a local ring, so
|w0|2 + |w1|2 = 1 = |z0|2 + |z1|2 implies |w0|2 ∈ U(R) or |w1|2 ∈ U(R)
and |z0|2 ∈ U(R) or |z1|2 ∈ U(R). Hence, w0 ∈ U(R) or w1 ∈ U(R) and
z0 ∈ U(R) or z1 ∈ U(R).
If z0 ∈ U(R) then we set λ = z−10 w0; if z1 ∈ U(R) then we set λ =
z−11 w1. Thus, λ ∈ S1(R) and (w0, w1) = (λz0, λz1). Because (z0, z1) ∈
162 Spheres over finite rings
S3(R) implies z0 ∈ U(R) or z1 ∈ U(R), we get h(R)−1(h(R)(z0, z1)) ∼=
S1(R).
(2) Given (q0, q1) ∈ S7(R), we follow mutatis mutandis (1) to complete
the proof.
�
By [1, Theorem 8.7], any commutative Artinian and unitary ring (in
particular, any finite commutative and unitary ring) is a finite prod-
uct of commutative Artinian local rings. Further, Sn(R1 × R2) ∼=
Sn(R1)×Sn(R2) for any commutative and unitary rings R1, R2 and n ≥ 0.
Consequently, in view of Proposition 3.8, for a commutative Artinian and
unitary ring R, and such that 2 is not a zero divisor in R, we get embed-
dings
h̄(R) : S3(R)/S1(R) −→ S2(R) and H̄(R) : S7(R)/S3(R) −→ S4(R).
In particular:
if R is a finite field with χ(R) 6= 2 then Corollary 2.3 and Theorem 3.2
imply that h̄(R) : S3(R)/S1(R) −→ S2(R) and H̄(R) : S7(R)/S3(R) −→
S4(R) are bijections;
if R = Zpk for an odd prime p and k ≥ 1 then Theorem 2.5 and
Proposition 3.7 lead to:
]S2(Zpk) ≥
{
p3k−2(p+ 1), if p ≡ 1 (mod 4);
p3k−2(p− 1), if p ≡ 3 (mod 4)
and
]S4(Zpk) ≥ p4k−2(p2 + 1).
Remark 3.9. Because
S15(R) ∼= {(c0, c1) ∈ R{e1, e2, e3, e4, e5, e6, e7}×R{e1, e2, e3, e4, e5, e6, e7};
|c0|2 + |c1|2 = 1},
we make use the Hopf map H : S15 → S8 to consider H(R) : S15(R)→
S8(R) for a commutative and unitary ring R, and state a result as in
Proposition 3.8 as well.
We close the paper with:
Conjecture 3.10. If p is an odd prime and k ≥ 1 then:
(1) ]S2(Zpk) =
{
p3k−2(p+ 1), if p ≡ 1 (mod 4);
p3k−2(p− 1), if p ≡ 3 (mod 4);
(2) ]S4(Zpk) = p4k−2(p2 + 1).
M. Golasiński, F. G. Ruiz 163
and
Problem 3.11. Let p be an odd prime and k ≥ 1. Find:
(1) ](Sn(Zpk)) for n > 4 with n 6= 7;
(2) the group structure of S3(Zpk).
References
[1] M.F. Atiyah and I.G. MacDonald, Introduction to commutative al-
gebra, Addison-Wesley Publishing Company, Reading, Massachusetts
(1969).
[2] Ch.W. Ayoub, On finite primary rings and their groups of units, Com-
pos. Math. vol. 21 (3), (1966), 247-252.
[3] J. Bochnak, On real algebraic morphisms into even-dimensional
spheres, Ann. of Math. 128 (1988), 415-433.
[4] J. Bochnak and W. Kucharz, Realization of homotopy classes by al-
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der Math. 36, Springer-Verlag, Berlin-Heidelberg-New York (1998).
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Sp × Sq, in Algebraic K-theory II, Lect. Notes in Math. 342 (1973),
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164 Spheres over finite rings
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Faculty of Mathematics and Computer Science
University of Warmia and Mazury
S loneczna 54, 10-710 Olsztyn, Poland
e-mail: marekg@matman.uwm.edu.pl
Departamento de Álgebra, Geometŕia y Topoloǵia
Facultad de Ciencias, Universidad de Málaga
Campus Universitario de Teatinos
29071 Málaga, España
e-mail: gomez_ruiz@uma.es
|
| id | oai:trim.imath.kiev.ua:article-295 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:09Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/b3/f880c010e47e860b10ebe1ae9ed618b3.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2952018-02-10T20:56:26Z Spheres over finite rings and their polynomial maps Сфери над скінченними кільцями та їх поліноміальні відображення Golasiński, M. Ruiz, F. G. Golasiński, M. Ruiz, F. G. The paper grew out of our attempt to describe all polynomial self-maps of the real andcomplex circle as well. Робота  виросла з нашої спроби описати всі поліноміальні відображення дійсного та комплексного кола в себе. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/295 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 148-164 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 148-164 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 148-164 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/295/284 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Golasiński, M. Ruiz, F. G. Golasiński, M. Ruiz, F. G. Spheres over finite rings and their polynomial maps |
| title | Spheres over finite rings and their polynomial maps |
| title_alt | Сфери над скінченними кільцями та їх поліноміальні відображення |
| title_full | Spheres over finite rings and their polynomial maps |
| title_fullStr | Spheres over finite rings and their polynomial maps |
| title_full_unstemmed | Spheres over finite rings and their polynomial maps |
| title_short | Spheres over finite rings and their polynomial maps |
| title_sort | spheres over finite rings and their polynomial maps |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/295 |
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