1001.0046.txt

arXiv:1001.0046v1  [math.CO]  31 Dec 2009THE CAUCHY-SCHWARZ INEQUALITY IN CAYLEY GRAPH
AND TOURNAMENT STRUCTURES ON FINITE FIELDS
STEPHAN FOLDES AND L ´ASZL´O MAJOR
Abstract. The Cayley graph construction provides a natural grid struc ture
on a finite vector space over a field of prime or prime square car dinality, where
the characteristic is congruent to 3 modulo 4, in addition to the quadratic
residue tournament structure on the prime subfield. Distanc e from the null
vector in the grid graph defines a Manhattan norm. The Hermiti an inner prod-
uct on these spaces over finite fields behaves in some respects similarly to the
real and complex case. An analogue of the Cauchy-Schwarz ine quality is valid
with respect to the Manhattan norm. With respect to the non-t ransitive order
provided by the quadratic residue tournament, an analogue o f the Cauchy-
Schwarz inequality holds in arbitrarily large neighborhoo ds of the null vector,
when the characteristic is an appropriate large prime.
1.Manhattan norms and grid graphs
We consider the finite fields FpandFp2of prime and prime square cardinality,
wherep≡3 mod 4. The field Fp2has a natural graph structure with the field
elements as vertices, two distinct vertices u,zbeing adjacent if ( z−u)4= 1. The
subfieldFpofFp2then induces a subgraph in which xandyare adjacent if and only
if (y−x)2= 1.The graph Fp2is isomorphic to the Cartesian square C2
p=Cp/squareCp,
whereCpis ap-cycle and within Fp2the induced subgraph Fpis itself a p-cycle.
Clearly the graph Fp2is not planar, but can be drawn as a grid on the torus.
For any connected graph whose vertex set is a group, the distanc e of any vertex
zfrom the identity element of the group is called the normofz, denoted N(z).
In general, distances and norms measured in connected subgraph s induced by sub-
groups can be larger than distances and norms measured with refe rence to the
whole graph. However, with respect to the distance-preserving s ubgraph induced
byFpinFp2, the norm of any z∈Fpis the same as its norm with respect to the
whole graph Fp2: this is simply the length of the shortest path from 0 to zin the
cycle induced by Fp.
Forq=porq=p2, then-dimensional vector space Fn
qis also endowed with
the Cartesian product graph structure Fq/square···/squareFqisomorphic to Cn
porC2n
p. The
norm of a vector v= (v1,...,v n) inFn
qis then equal to the sum N(v1)+···+N(vn)
and we also write N(v) for this vector norm.
The Gaussian integers Z[i] also constitute a graph in which uandzare adjacent
if and only if ( z−u)4= 1.
Date: Dec 24, 2009.
1991Mathematics Subject Classification. Primary 05C12, 05C20, 05C25; Secondary 06F99,
11T99.
Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, submult iplicativity,
finite field, quadratic field extension, quadratic residue to urnament, grid graph, Manhattan dis-
tance, discrete norm, Gaussian integers, graph product, gr aph quotient, Cayley graph.
12 STEPHAN FOLDES AND L ´ASZL´O MAJOR
It iseasytoseethatthenorminthis infinite Manhattan grid satisfiesthetriangle
and submultiplicative inequalities
N(u+z)≤N(u)+N(z)
N(uz)≤N(u)N(z)
To emphasize that the norms on Fp2,Fn
p2andZ[i] are understood with reference
to the specific grid graphs defined above, we call these norms Manhattan norms .
Throughout this paper we think of Fp2as the ring quotient Z[i]/(p).
2.Graph quotients and Cayley graphs
Given a graph G(undirected, with possible loops) on vertex set Vand an equi-
valence relation ≡onV, thequotient graph G/≡is defined as follows: the vertices
ofG/≡are the equivalence classes of ≡, and classes A,Bare adjacent if for some
a∈A,b∈B, the elements a,bare adjacent in G. Note that the distance of Ato
Bin the quotient graph is at most equal to, but possibly less than the m inimum of
the distances atobfor alla∈A,b∈B. Note also that G/≡can have loops even
ifGhas not.
Given a group Gwith identity element eand a set Γ of group elements that
generates G, the(left) Cayley graph C(G,Γ) ofGwith respect to Γ has vertex set
G, elements a,b∈Gbeing considered adjacent if ab−1orba−1belongs to Γ. For
each congruence ≡of the group G, corresponding to some normal subgroup H, Γ
yields a generating set Γ ≡ofG/≡consisting with those classes of ≡that intersect
Γ. The graph quotient of C(G,Γ) by the equivalence ≡coincides with the Cayley
graph of the quotient graph G/≡with respect to Γ ≡. ForR⊆Ginducing a
connected subgraph [ R] inC(G,Γ), denote by dR(x,y) the distance function of the
subgraph [ R]. Denoting by xHtheH-coset of any x∈G, this relates to norms in
C(G,Γ) andC(G,Γ)/≡as follows: for all x∈R,
dR(x,e)≥N(x)≥N(xH)
Both inequalities can be strict. However, we have:
Cayley Graph Quotient Lemma. Let a group Gwith identity ebe generated
byΓ⊆G, and consider any normal subgroup Hwith corresponding congruence ≡.
There is a set R⊆Ghaving exactly one element in common with each congruence
class modulo H, and such that for every x∈R
dR(x,e) =N(x) =N(xH)
Proof.We can define the unique (representative) element r(A)∈R∩Afor each
cosetAby induction on the distance d(H,A) ofAfromHinC(G,Γ)/≡. Let
r(H) =e. Assuming r(A) defined for all Awithd(H,A)≤m, let a coset Bhave
distance m+1 from H. Choose any coset Aadjacent to Bwithd(H,A) =mand
elements a∈A,b∈Bthat are adjacent in C(G,Γ). Letr(B) =ba−1r(A)./square
We can apply the above lemma in the case where G=Z[i], Γ ={1,i}and
H=pZ[i] ={pa+pbi:a,b∈Z}foraprimeinteger p≡3 mod 4. Now C(G,Γ)and
C(G,Γ)/≡are the Manhattan grid graphs on Z[i] andZ[i]/H=Fp2, respectively.
Referringtothe set Rofrepresentativesinthe lemma, forany H-cosetsX,Yletx,y
be the unique elements in X∩R,Y∩R. Asxy∈XY, we have N(XY)≤N(xy).
By the submultiplicative inequality in Z[i] we have N(xy)≤N(x)N(y). Using the3
lemmawehave N(x)N(y) =N(X)N(Y). Thisyieldsasubmultiplicativeinequality
inFp2and a similar reasoning on the coset X+Yyields a triangle inequality:
Triangle and Submultiplicative Inequalities in Fp2.For allu,zinFp2
N(u+z)≤N(u)+N(z)
N(uz)≤N(u)N(z)
/square
This indicates that Manhattan distance provides a well-behaved not ion of neigh-
borhood of 0 in the finite fields Fp2.
3.Squares in Fpand non-transitive order
For each prime p≡3 mod 4 the quadratic residue tournament onFpis the
directed graph with vertex set Fpin which there is an arrowfrom vertex xto vertex
yify−xis a non-zero square in Fp, in which case we write x <py. We write x≤py
ifx <pyorx=y. The relation ≤pis reflexive, anti-symmetric but not transitive,
and for every x/ne}ationslash=yexactly one of x≤pyory≤pxholds. Using Dirichlet’s
theorem on primes in arithmetic progressions, Kustaanheimo showe d [4] that for
every positive integer k, there is a prime p≡3 mod 4, such that ≤pis a transitive
(and linear) order relation on {0,1,...,k} ⊆Fp, that is, all positive integers up to
kare quadratic residues mod p. Obviously kcannot exceed ( p−1)/2. Implications
of [4] and related questions were investigated by J¨ arnefelt, Kust aanheimo, Quist
[3, 5], in particular with a view to discrete models in physics, also in subse quent
application-oriented work between the 1950’s (Coish [1]) and the 198 0’s (Nambu
[6]). For further references see [2]. In particular [4] implies that for every positive
integerk, there is a prime p≡3 mod 4, such that all z∈Fp2withN(z)≤kare
squares in Fp2. (Note that all elements of the prime subfield Fpare squares in Fp2.)
To emphasise the analogy of the relation ≤pwith the ordinary inequality relation
≤among numbers, we say that a non-zero z∈Fp2ispositiveifz∈Fpand 0≤pz.
4.Inner products compared in non-transitive order
The only non-trivial automorphism of the field Fp2associates to each z∈Fp2
itsconjugate z. Theinner product v·wof vectors v= (v1,...,v n) andw=
(w1,...,w n) inFn
p2is defined as the scalar v1w1+···+vnwn∈Fp2. This inner
product is left and right distributive over vector addition, satisfies v·w=w·v,
c(v·w) = (cv)·w=v·(cw) for allc∈Fp2. However, while v·vbelongs to the
prime subfield Fp,v·vis not necessarily positive, and can be 0 even if v/ne}ationslash=0. Still,
a conditional version of positive definiteness holds locally:
Conditional Positive Definiteness. For every k≥1there is a prime p≡3
mod 4, such that for all n≥1and for all vectors v∈Fn
p2of Manhattan norm
N(v)≤k,we have 0≤pv·vwith equality if and only if v=0.
Proof.By Kustaanheimo’s result in [4] there is a prime integer p≡3 mod 4 such
that 0,1,...,2k3are all quadratic residues mod p. Forv= (v1,...,v n) inFn
p2, let
vj=aj+bji, wherei2=−1. IfN(v)≤kthen for all j,N(aj)≤kandN(bj)≤k,
vjvj=a2
j+b2
jbelongs to the set of squares {0,...,2k2}. Sincevjcan be non-zero
for at most kindices 1 ≤j≤nonly, the sum of the corresponding terms a2
j+b2
j
belongs to the set of squares {0,1,...,2k3}. /square4 STEPHAN FOLDES AND L ´ASZL´O MAJOR
Note that for all vectors v,w∈Fn
p2
(v·w)(w·v) = (v·w)(v·w)∈Fpand
(v·v)(w·w)∈Fp
Ifvandwareproportional , i.e. if there exists a scalar cinFp2such that v=cw
orw=cv, then the above two products are equal. Generally, they are relat ed in
the quadratic residue tournament of Fpas follows.
Cauchy-Schwarz Inequality for Quadratic Residue Tourname nts.For eve-
ryk≥1there is a prime p≡3 mod 4 , such that for all n≥1and for all vectors
v,w∈Fn
p2of Manhattan norm at most k,
(v·w)(w·v)≤p(v·v)(w·w)
Proof.Forn= 1 the inequality holds trivially as the two sides are equal. Assume
n≥2,v= (v1,...,v n),w= (w1,...,w n). Forall1 ≤i≤n,N(vi)≤k,N(wi)≤k.
By Kustaanheimo’s result [4] there is a prime p≡3 mod 4 such that all positive
integers up to 4 k6are quadratic residues modulo p. For each of the/parenleftbign
2/parenrightbig
pairs
{i,j} ⊆ {1,...,n},i/ne}ationslash=j, by the triangle and submultiplicative inequalities in Fp2
N[(viwj−vjwi)(viwj−vjwj)]≤(k2+k2)2= 4k4
Thus the element
(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj) = (viwj−vjwi)(viwj−vjwj)
is a square of Manhattan norm at most 4 k4inFp, and it is non-zero for at most/parenleftbigk
2/parenrightbig
≤k2pairs{i,j}. Summing over all pairs {i,j}, all but at most/parenleftbigk
2/parenrightbig
≤k2terms
vanish in the sum/summationdisplay
[(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj)]
which therefore has Manhattan norm at most 4 k6and it must also be a square in
Fp. But this sum is equal to the difference of products
n/summationdisplay
i=1vivin/summationdisplay
j=1wjwj−n/summationdisplay
i=1viwin/summationdisplay
j=1vjwj= (v·v)(w·w)−(v·w)(w·v)
which is consequently a square in Fp. /square
Remark. From the proof it is clear that, in analogy with the classical Cauchy-
Schwarz inequality, for vectors v,wof norm not exceeding kinFn
p2, wherepis
related to kas stipulated above, the Cauchy-Schwarz inequality with respect t o≤p
holds with equality if and only if viwj−vjwi= 0 for all i,j, i.e. if and only if v,w
are proportional.
We note that the inequality established above is conditional, it holds on ly in a
specified Manhattan neighborhood of the null vector. Every non- zero element of
Fpcan be written as a sum of two squares, in particular there are a,b∈Fp, such
thata2+b2=−1. Forz=a+biwe have zz=−1. As soon as n≥2, inFn
p2let
v= (a,b,0,...,0) and w= (bz,−az,0,...,0)
The inequality ( v·w)(w·v)≤p(v·v)(w·w) fails because the left-hand side is 0
and the right-hand side is −1. In fact if n≥3, the inequality can be invalidated
with vectors v,winFn
pas follows. Taking again a,b∈Fpwitha2+b2=−1, let
v= (1,a,b,0,...,0) and w= (1,0,0,0,...,0)5
However,the Cauchy-Schwarzinequalityholdsunconditionallyinthe 2-dimensional
case for vectors with components in Fp:
Special case of F2
p.Letpbe a prime congruent 3modulo4. For all vectors v,w
inF2
p
(v·w)(w·v)≤p(v·v)(w·w)
Proof.Nowtheconjugationappearinginthe innerproductsisthe identity. Written
in components,
(v·v)(w·w)−(v·w)(w·v) = (v2
1+v2
2)(w2
1+w2
2)−(v1w1+v2w2)2=
=v2
1w2
2+v2
2w2
1−2v1w1v2w2= (v1w2−v2w1)2
/square
5.Manhattan norm of inner product
The Manhattan norm can be seen to be submultiplicative not only on th e ring
Z[i] and its quotient field Fp2, but on all vector spaces Fn
p2, with respect to the
inner product:
Cauchy-Schwarz Inequality for Manhattan Norm on Fn
p2.Consider any
primep≡3 mod 4 and letn≥1. For all v,w∈Fn
p2
N(v·w)≤N(v)N(w)
Proof.Letv= (v1,...,v n),w= (w1,...,w n)∈Fn
p2. Thenv·w=/summationtextvjwj. Clearly
N(z) =N(z) for any z∈Fp2. By the triangle and submultiplicative inequalities in
Fp2we have
N(v·w) =N/parenleftbig/summationtextvjwj/parenrightbig
≤/summationtextN/parenleftbig
vjwj/parenrightbig
≤/summationtextN(vj)N(wj)≤
≤/summationtextN(vj)/summationtextN(wj) =N(v)N(w)
/square
Remark. The inequality N(v·w)≤N(v)N(w) is easily interpreted and continues
to hold for v,win the module ( Z[i]/mZ[i])nfor any positive integer m. As soon as
mis composite, or a prime not congruent to 3 modulo 4, the ring Z[i]/mZ[i] fails
to be an integral domain.
References
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[2] S. Foldes, The Lorentz group and its finite field analogues : local isomorphism and approxima-
tion, J. Math. Phys. 49, 093512 (2008)
[3] G. J¨ arnefelt, P. Kustaanheimo, An observation on finite geometries, in Proc. Skandinaviske
Matematikerkongress i Trondheim 1949, 166-182
[4] P. Kustaanheimo, A note on a finite approximation of the eu clidean plane geometry, Comment.
Phys.-Math. Soc. Sc. Fenn. XV. 19 (1950) 1-11
[5] P. Kustaanheimo, B. Qvist, On differentiation in Galois fi elds. Ann. Acad. Sci. Fennicae. Ser.
A. I. Math.-Phys. 1952, (1952). no. 137, 12 pp.
[6] Y. Nambu, Field theory of Galois fields, in Field Theory an d Quantum Statistics, eds. J.A.
Batalin et.al., Institute of Physics Publishing 1987, pp. 6 25-6366 STEPHAN FOLDES AND L ´ASZL´O MAJOR
Stephan Foldes
Institute of Mathematics,
Tampere University of Technology,
PL 553, 33101 Tampere, Finland
E-mail address :[email protected]
L´aszl´o Major
Institute of Mathematics,
Tampere University of Technology,
PL 553, 33101 Tampere, Finland
E-mail address :[email protected]