A Generalization of the Doubling Construction for Sums of Squares Identities
The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple [r,s,m] a series of new ones [r+ρ(2ⁿ⁻¹),2ⁿ...
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2017
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| Цитувати: | A Generalization of the Doubling Construction for Sums of Squares Identities / C. Zhang, H.L. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1487562019-02-24T15:44:44Z A Generalization of the Doubling Construction for Sums of Squares Identities Zhang, C. Huang, H.L. The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple [r,s,m] a series of new ones [r+ρ(2ⁿ⁻¹),2ⁿs,2ⁿm] for all positive integer n, where ρ is the Hurwitz-Radon function. 2017 Article A Generalization of the Doubling Construction for Sums of Squares Identities / C. Zhang, H.L. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11E25 DOI:10.3842/SIGMA.2017.064 http://dspace.nbuv.gov.ua/handle/123456789/148756 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple [r,s,m] a series of new ones [r+ρ(2ⁿ⁻¹),2ⁿs,2ⁿm] for all positive integer n, where ρ is the Hurwitz-Radon function. |
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Article |
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Zhang, C. Huang, H.L. |
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Zhang, C. Huang, H.L. A Generalization of the Doubling Construction for Sums of Squares Identities Symmetry, Integrability and Geometry: Methods and Applications |
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Zhang, C. Huang, H.L. |
| author_sort |
Zhang, C. |
| title |
A Generalization of the Doubling Construction for Sums of Squares Identities |
| title_short |
A Generalization of the Doubling Construction for Sums of Squares Identities |
| title_full |
A Generalization of the Doubling Construction for Sums of Squares Identities |
| title_fullStr |
A Generalization of the Doubling Construction for Sums of Squares Identities |
| title_full_unstemmed |
A Generalization of the Doubling Construction for Sums of Squares Identities |
| title_sort |
generalization of the doubling construction for sums of squares identities |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/148756 |
| citation_txt |
A Generalization of the Doubling Construction for Sums of Squares Identities / C. Zhang, H.L. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 9 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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| first_indexed |
2025-07-12T20:10:16Z |
| last_indexed |
2025-07-12T20:10:16Z |
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1837473227291492352 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 064, 6 pages
A Generalization of the Doubling Construction
for Sums of Squares Identities
Chi ZHANG † and Hua-Lin HUANG ‡
† School of Mathematics, Shandong University, Jinan 250100, China
E-mail: chizhang@mail.sdu.edu.cn
‡ School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
E-mail: hualin.huang@hqu.edu.cn
Received May 16, 2017, in final form August 13, 2017; Published online August 16, 2017
https://doi.org/10.3842/SIGMA.2017.064
Abstract. The doubling construction is a fast and important way to generate new solutions
to the Hurwitz problem on sums of squares identities from any known ones. In this short
note, we generalize the doubling construction and obtain from any given admissible triple
[r, s, n] a series of new ones [r+ ρ(2m−1), 2ms, 2mn] for all positive integer m, where ρ is the
Hurwitz–Radon function.
Key words: Hurwitz problem; square identity
2010 Mathematics Subject Classification: 11E25
1 Introduction
In his seminal paper [2], Hurwitz addressed the famous problem: Determine all the sums of
squares identities(
x21 + x22 + · · ·+ x2r
)(
y21 + y22 + · · ·+ y2s
)
= z21 + z22 + · · ·+ z2n, (1.1)
where X = (x1, x2, . . . , xr) and Y = (y1, y2, . . . , ys) are systems of indeterminants and every zk
is a bilinear form of X and Y with coefficients in some given field. If there does exist such an
identity, we call [r, s, n] an admissible triple. This problem of Hurwitz has close connections to
various topics in algebra, arithmetic, combinatorics, geometry, topology, etc. Many mathemati-
cians have studied this Hurwitz problem during the past century. See [7] for an overview.
Though a complete solution to the Hurwitz problem is still far out of reach at present, many
admissible triples have been obtained in the literature. In particular, the admissible triples of
form [r, n, n] was settled independently by Hurwitz in [3] and by Radon in [6]. The celebrated
Hurwitz–Radon theorem states that [r, n, n] is admissible if and only if r ≤ ρ(n) where ρ is the
Hurwitz–Radon function defined by ρ(n) = 8α + 2β if n = 24α+β(2γ + 1) with 0 ≤ β ≤ 3. In
the early 1980s, Yuzvinsky introduced the novel idea of orthogonal pairings [8] and proposed
in [9] the following three families of admissible triples in the neighborhood of the Hurwitz–Radon
triples
[
2n+ 2, 2n − ϕ(n), 2n
]
, where ϕ(n) =
(
n
n/2
)
, n ≡ 0 mod 4,
2
(
n−1
(n−1)/2
)
, n ≡ 1 mod 4,
4
(
n−2
(n−2)/2
)
, n ≡ 2 mod 4.
(1.2)
The first two families are confirmed in [4] and the third in [1]. Moreover, some new families of
admissible triples are constructed in [1, 5].
mailto:chizhang@mail.sdu.edu.cn
mailto:hualin.huang@hqu.edu.cn
https://doi.org/10.3842/SIGMA.2017.064
2 C. Zhang and H.-L. Huang
Another natural idea of constructing admissible triples is to find some general procedures to
generate new ones from known ones. Among which, the doubling construction, i.e., generating
an admissible triple [r+1, 2s, 2n] from any given triple [r, s, n], is a fast and important program.
In the present note we consider arbitrarily iterated doubling constructions and the aim is to
optimize the obvious triples [r +m, 2ms, 2mn]. The well-known Hurwitz–Radon triples suggest
that the first item might be properly increased as the form given below via the function ρ.
Main Theorem. If [r, s, n] is admissible, then so is [r + ρ(2m−1), 2ms, 2mn] for all positive
integer m.
Though our observation arises from the idea used in [1], it turns out that a more elementary
approach of matrices will suffice for a proof.
2 Proof of the main theorem
First, we introduce the so-called admissible matrices to reformulate the Hurwitz problem. Then,
as a trial we provide a proof via admissible matrices for the classical doubling construction.
Finally, we extend the idea to iterated doubling constructions and complete the proof for the
main theorem.
2.1 Admissible matrices
The notion of admissible matrices arises naturally from an attempt to reformulate the sums of
squares identity (1.1) by a system of polynomial equations. Indeed, in (1.1) if we write
zk =
∑
1≤i≤r
1≤j≤s
ci,j,kxiyj ,
then it is easy to see that the identity (1.1) is equivalent to the following system of algebraic
equations
n∑
k=1
c2i,j,k = 1, 1 ≤ i ≤ r, 1 ≤ j ≤ s,
n∑
k=1
ci1,j,kci2,j,k = 0, 1 ≤ i1 < i2 ≤ r, 1 ≤ j ≤ s,
n∑
k=1
ci,j1,kci,j2,k = 0, 1 ≤ i ≤ r, 1 ≤ j1 < j2 ≤ s,
n∑
k=1
(ci1,j1,kci2,j2,k + ci1,j2,kci2,j1,k) = 0, 1 ≤ i1 < i2 ≤ r, 1 ≤ j1 < j2 ≤ s. (2.1)
In the rest of the note, we always regard the resulting cuboid A := (ci,j,k)r×s×n as an r×s matrix
with (i, j)-entry the n-dimensional vector Ai,j := (ci,j,1, ci,j,2, . . . , ci,j,n). Taking the formal inner
product on n-dimensional vectors, namely 〈(u1, u2, . . . , un), (v1, v2, . . . , vn)〉 := u1v1 + u2v2 +
· · ·+ unvn, then (2.1) can be rewritten as
(1) 〈Ai,j , Ai,j〉 = 1, 1 ≤ i ≤ r, 1 ≤ j ≤ s,
(2) 〈Ai1,j , Ai2,j〉 = 0, 1 ≤ i1 < i2 ≤ r, 1 ≤ j ≤ s,
(3) 〈Ai,j1 , Ai,j2〉 = 0, 1 ≤ i ≤ r, 1 ≤ j1 < j2 ≤ s,
(4) 〈Ai1,j1 , Ai2,j2〉+ 〈Ai1,j2 , Ai2,j1〉 = 0, 1 ≤ i1 < i2 ≤ r, 1 ≤ j1 < j2 ≤ s. (2.2)
Obviously, the existence of such a matrix A is equivalent to the existence of an admissible triple
of size [r, s, n]. In keeping the terminologies coherent, such matrices are said to be admissible.
A Generalization of the Doubling Construction for Sums of Squares Identities 3
2.2 The doubling construction revisited
For a better explanation of our method, firstly we provide a proof by admissible matrices for the
classical doubling construction. Some preparing definitions and notations are necessary. Let k
be a field of characteristic not 2.
Definition 2.1. Fix two integers n and m. A vector in α = (α1, α2, . . . , αmn) ∈ kmn is said to
be in level k ∈ {1, 2, . . . ,m} if its arguments αl are 0 unless (k − 1)n+ 1 ≤ l ≤ kn. Let β ∈ kn.
We call γ ∈ kmn a positive copy of β in level k if γ(k−1)n+i = βi (1 ≤ i ≤ n) and other arguments
of γ are 0. Similarly, we call γ a negative copy of β in level k if γ(k−1)n+i = −βi (1 ≤ i ≤ n) and
other arguments of γ are 0.
Remark 2.2. Let α1, α2 ∈ kmn and β1, β2 ∈ kn.
1. If α1 is a copy of β1 in level k, α2 is a copy of β2 in level l and they have the same sign,
then 〈α1, α2〉 = 〈β1, β2〉 if k = l, 〈α1, α2〉 = 0 if k 6= l.
2. If α1 is a copy of β1 in level k, α2 is a copy of β2 in level l and they have different signs,
then 〈α1, α2〉 = −〈β1, β2〉 if k = l, 〈α1, α2〉 = 0 if k 6= l.
Given an admissible triple of size [r, s, n], we have a corresponding r×s admissible matrix A.
We shall construct an (r+1)×2s admissible matrix B whose entries are 2n-dimensional vectors
as follows:
1) Bi,j is a positive copy of Ai,j for 1 ≤ i ≤ r, 1 ≤ j ≤ s in level 1,
2) Bi,s+j is a positive copy of Ai,j for 2 ≤ i ≤ r, 1 ≤ j ≤ s in level 2,
3) Br+1,j is a positive copy of A1,j for 1 ≤ j ≤ s in level 2,
4) Br+1,s+j is a positive copy of A1,j for 1 ≤ j ≤ s in level 1,
5) B1,s+j is a negative copy of A1,j for 1 ≤ j ≤ s in level 2.
We give a detailed verification of the admissibility of B and hope this will shed some light
on the study of iterated doubling constructions.
1. Every Bi,j is a copy of some entry Ak,l of A , so 〈Bi,j , Bi,j〉 = 〈Ak,l, Ak,l〉 = 1, hence (1)
of (2.2) holds.
2. 〈Bi,j1 , Bi,j2〉 = 0 (j1 < j2). Indeed, if 1 ≤ j1 ≤ s and s+1 ≤ j2 ≤ 2s, then the two vectors
are in different levels; if 1 ≤ i ≤ r + 1, 1 ≤ j1 < j2 ≤ s, 〈Bi,j1 , Bi,j2〉 = 〈Ai,j1 , Ai,j2〉 = 0.
A similar argument works for s+1 ≤ j1 < j2 ≤ 2s. So (2) of (2.2) holds. In the same way,
one can show that (3) of (2.2) holds.
3. For (4) of (2.2), we need to verify 〈Bi1,j1 , Bi2,j2〉+ 〈Bi1,j2 , Bi2,j1〉 = 0 (i1 < i2, j1 < j2).
(a) If 1 ≤ j1 ≤ s < j2 ≤ 2s and 1 ≤ i1 < i2 ≤ r, Bi1,j1 and Bi2,j1 are in level 1 and Bi1,j2
and Bi2,j2 are in level 2. Hence the equation is obvious.
(b) If 1 ≤ j1 ≤ s, s+ 1 ≤ j2 ≤ 2s and 1 ≤ i1 ≤ r, i2 = r+ 1, Bi1,j1 and Bi2,j2 are in level
1 and Bi1,j2 and Bi2,j1 are in level 2. 〈Bi1,j1 , Bi2,j2〉+ 〈Bi1,j2 , Bi2,j1〉 = 〈Ai1,j1 , A1,j2〉+
〈Ai1,j2 , A1,j1〉 = 0. If i1 = 1 and j2 = j1 + s, then 〈Bi1,j1 , Bi2,j2〉 + 〈Bi1,j2 , Bi2,j1〉 =
〈A1,j1 , A1,j1〉 − 〈A1,j1 , A1,j1〉 = 1− 1 = 0.
(c) If 1 ≤ i1 < i2 ≤ r, 1 ≤ j1 < j2 ≤ s or s + 1 ≤ j1 < j2 ≤ 2s, the four vectors are in
the same level. If i1 = 1, s+ 1 ≤ j1 ≤ j2 ≤ 2s, then 〈Bi1,j1 , Bi2,j2〉+ 〈Bi1,j2 , Bi2,j1〉 =
−〈Ai1,j1 , Ai2,j2〉 − 〈Ai1,j2 , Ai2,j1〉 = 0. Otherwise, 〈Bi1,j1 , Bi2,j2〉 + 〈Bi1,j2 , Bi2,j1〉 =
〈Ai1,j1 , Ai2,j2〉+ 〈Ai1,j2 , Ai2,j1〉 = 0.
4 C. Zhang and H.-L. Huang
(d) If 1 ≤ i1 ≤ r, i2 = r + 1, 1 ≤ j1 < j2 ≤ s or s + 1 ≤ j1 < j2 ≤ 2s, then Bi1,j1
and Bi2,j2 are in different levels and this is also the case for Bi1,j2 and Bi2,j1 . Now
the equation is clear.
Thus, (4) of (2.2) holds.
Remark 2.3. The matrix B used in the above proof can be illustrated by the following table:
1 2
1 2
2 1
Here, cells in the first and the third rows are copies of the first row of A, and cells in the second
rows are copies of the submatrix of A obtained by deleting the first row. The number given in
the center of a cell represents the level of the vectors therein. The sign of a cell is indicated by
its color: white means positive, gray means negative.
Above all, the table provides a visual admissibility of B. The conditions (1)–(3) of (2.2) are
immediate, as the vectors are either in different levels, or essentially can be considered within A.
The same reasoning also works for (4) of (2.2) in most cases. As for the case i1 = 1, i2 = r + 1,
j1 + s = j2, one further needs to take the signs into consideration. In fact, this also tells us
in the very beginning how to manipulate the signs of the copies of cells so that (4) holds. Of
course, the signing is far from unique. Just for such B, we have 16 kinds of correct schemes as
follows. These tables are useful in the following for the verification of the admissibility of bigger
matrices.
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
1 2
1 2
2 1
A Generalization of the Doubling Construction for Sums of Squares Identities 5
2.3 Iterated doubling constructions
Now we are ready to prove the main theorem. As before, let A be an admissible matrix corre-
sponding to an admissible triple of size [r, s, n]. We will provide admissible matrices in terms of
tables as in Remark 2.3 which will induce admissible triples of sizes [r+2, 4s, 4n], [r+4, 8s, 8n]
and [r + 8, 16s, 16n].
1 2 3 4
1 2 3 4
2 1 4 3
3 4 1 2
the table of [r + 2, 4s, 4n]
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
2 1 4 3 6 5 8 7
3 4 1 2 7 8 5 6
5 6 7 8 1 2 3 4
8 7 6 5 4 3 2 1
the table of [r + 4, 8s, 8n]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15
3 4 1 2 7 8 5 6 11 12 9 10 15 16 13 14
5 6 7 8 1 2 3 4 13 14 15 16 9 10 11 12
8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9
9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
12 11 10 9 16 15 14 13 4 3 2 1 8 7 6 5
14 13 16 15 10 9 12 11 6 5 8 7 2 1 4 3
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2
the table of [r + 8, 16s, 16n]
As the table of [r + 4, 8s, 8n] and the table of [r + 2, 4s, 4n] are both subtables of that of
[r + 8, 16s, 16n], we just explain the last table:
1. Every cell in the first row of a table which stands for the first row of the corresponding
admissible matrix is the copy of the first row of A.
2. Every cell in the second row of a table which stands for the rows from second to r-th of
the corresponding admissible matrix is the copy of the rows from second to r-th of A.
3. Every cell in other rows of a table which stand for the k-th rows (k ≥ r + 1) of the
corresponding admissible matrix is the copy of the first row of A.
4. For every cell, the number means the levels and the color means the signs.
Using the same discussion of Remark 2.3, it is easy to verify that (1)–(3) of (2.2) hold. For the
verification of (4) of (2.2), it is enough to consider the 8 added rows and verify those entries
which are in the same level. Then the admissibility follows by a direct and simple computation.
6 C. Zhang and H.-L. Huang
Acknowledgements
This research was supported by NSFC 11471186 and NSFC 11571199.
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https://doi.org/10.1080/03081088408817582
1 Introduction
2 Proof of the main theorem
2.1 Admissible matrices
2.2 The doubling construction revisited
2.3 Iterated doubling constructions
References
|