Incidence bounds and applications over finite fields

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by  Nguyen Duy Phuong

Incidence bounds and applications over finite fields
Nguyen Duy Phuong Thang Pham Nguyen Minh Sang
∗ † ‡
6
Claudiu Valculescu Le Anh Vinh
1 § ¶
0
2

n
a
J Abstract

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In this paper we introduce a unified approach to deal with incidence problems

]
between points and varieties over finite fields. More precisely, we prove that the
O
number of incidences I( , ) between a set of points and a set of varieties of
C
P V P V
a certain form satisfies
.
h
t
a I( , ) |P||V| qdk/2 .
m P V − qk ≤ |P||V|
(cid:12) (cid:12)
[ (cid:12) (cid:12) p
Thisresultisagenera(cid:12)(cid:12)lization oftheres(cid:12)(cid:12)ultsofVinh(2011), Bennettetal. (2014),
1
and Cilleruelo et al. (2015). As applications of our incidence bounds, we obtain
v
0 results on the pinned value problem and the Beck type theorem for points and
9
spheres.
2
Usingtheapproach introduced, wealso obtain aresultonthenumberof distinct
0
0 distancesbetweenpointsandlinesinF2,whichisthefinitefieldanalogousofarecent
q
.
1 result of Sharir et al. (2015).
0
6
1
1 Introduction
:
v
i
X
In 1983, Szemer´edi and Trotter [30] proved that for any set of n points, and any set
r P L
a of n lines in the plane, the number of incidences between points of and lines from is
P L
asymptotically at most n4/3. Apart from being interesting in itself and being a useful tool
for various other discrete mathematics problem, the Szemer´edi-Trotter theorem allowed
various exentsions and generalizations. T´oth proved that the same bound holds when we
work over the complex plane (see [31] for more details). Pach and Sharir [25] generalized
the Szemer´edi-Trotter theorem to the case of points and curves [25].
∗Vietnam National University, Email: [email protected]
†EPFL, Lausanne, Switzerland. Research partially supported by Swiss National Science Foundation
Grants 200020-144531and 200021-137574. Email: [email protected]
‡Vietnam National University, Email: [email protected]
§EPFL, Lausanne, Switzerland. Research partially supported by Swiss National Science Foundation
Grants 200020-144531and 200021-137574. Email: [email protected]
¶Vietnam National University, Email: [email protected]
1
Let F be a finite field of q elements where q is an odd prime power. Let be a set of
q
P
points and be a set of lines in F2, and I( , ) be the number of incidences between
L q P L P
and . In [3], Bourgain, Katz, and Tao proved that if one has N lines and N points in
L
the plane F2 for some 1 N q2, then there are at most O(N3/2−ǫ) incidences. Here
q ≪ ≪
and throughout, X & Y means that X CY for some constant C and X Y means
≥ ≫
that Y = o(X), where X,Y are viewed as functions of the parameter q. The study of
incidence problems over finite fields received a considerable amount of attention in recent
years [5, 9, 16, 20, 21, 26, 28, 23, 32, 33, 34].
Note that the bound N3/2 can be easily obtained from extremal graph theory. The
relation between ǫ and α in the result of Bourgain, Katz, and Tao is difficult to determine,
and it is far from being tight. If N = log log log q 1, then Grosu [9] proved that one
2 6 18 −
can embed the point set and the line set to C2 without changing the incidence structure.
Thus it follows from a tight bound on the number of incidences between points and lines
in C2 due to T´oth [31] that I( , ) = O(N4/3). By using methods from spectral graph
P L
theory, the fifth listed author [33] proved the tight bound for the case N q as follows.
≫
Theorem 1.1 (Vinh, [33]). Let be a set of points and be a set of lines in F2. Then
P L q
we have
I( , ) |P||L| q1/2 . (1.1)
P L − q ≤ |P||L|
(cid:12) (cid:12)
(cid:12) (cid:12) p
It follows from Theorem 1(cid:12).1 that when N (cid:12) q3/2, the number of incidences between
(cid:12) (cid:12)
≥ P
and is asymptotically at most (1+o(1))N4/3 (this meets the Szemer´edi-Trotter bound).
L
Furthermore, if q3, then the number of incidences is close to the expected value
|P||L| ≫
/q. The lower bound in the theorem is also proved to be sharp up to a constant
|P||L|
factor, in the sense that there is a set of points and a set of lines with = = q3/2
P L |P| |L|
that determines no incidence (for details see [34]). Theorem 1.1 has various applications
in several combinatorial number theory problems (for example [13, 14, 19]).
The main purpose of this paper is to introduce a unified approach, which allows us
to deal with incidence problems between points and certain families of varieties. As
applications of incidence bounds, we obtain results on the pinned value problem and the
Beck type theorem for points and spheres. Using this approach, we also obtain a result
on the number of distinct distances between points and lines in F2.
q
1.1 Incidences between points and varieties
We first need the following definitions.
Definition 1.2. Let S be a set of polynomials in F [x ,...,x ]. The variety determined
q 1 d
by S is defined as follows
V(S) := p Fd : f(p) = 0 for all f S .
{ ∈ q ∈ }
Let h (x),...,h (x) be fixed polynomials of degree at most q 1 in F [x ,...,x ], and
1 k q 1 d
−
let b = (b ,...,b ), with 1 i k, be fixed vectors in (Z+)d, and gcd(b ,q 1) = 1
i i1 id ij
≤ ≤ −
2
for all 1 j d. For any k-tuple (a ,...,a ) with a = (a ,...,a ,a ) Fd+1, we
≤ ≤ 1 k i i1 id i(d+1) ∈ q
define
d
f (x,a ) := h (x)+a xbi, where a xbi := a xbij +a .
i i i i · i · ij j i(d+1)
j=1
X
Also, we define the corresponding families of varieties as follows:
V := V (x f (x,a ),...,x f (x,a )) Fd+k, and
a1,...,ak d+1 − 1 1 d+k − k k ⊆ q
W := V (f (x,a ),...,f (x,a )) Fd.
a1,...,ak 1 1 k k ⊆ q
Similarly to incidences between points of lines, given a set of points and a set of
P V
varieties, we define the number of incidences I( , ) between and as the cardinality
P V P V
of the set (p,v) p v . Our first main result is as follows:
{ ∈ P ×V | ∈ }
Theorem 1.3. Let be a set of points in Fd Fk and a set of varieties of the form
P q × q V
V defined above. Then the number of incidences between and satisfies
a1,...,ak P V
I( , ) |P||V| qdk/2 .
P V − qk ≤ |P||V|
(cid:12) (cid:12)
(cid:12) (cid:12) p
(cid:12) (cid:12)
As a consequence of Theorem 1.3, we obtain the following result.
(cid:12) (cid:12)
Corollary 1.4. Let be a set of points in Fd and be a set of varieties of the form
P q V
W defined above. Then the number of incidences between and satisfies
a1,...,ak P V
I( , ) |P||V| qdk/2 .
P V − qk ≤ |P||V|
(cid:12) (cid:12)
(cid:12) (cid:12) p
(cid:12) (cid:12)
Let us observe that if hi(cid:12)(x) 0 and bi =(cid:12) (1,...,1) for all 1 i k, then a variety
≡ ≤ ≤
of the form V is a k-flat in the vector space Fd+k. Therefore, we recover the bound
a1,...,ak q
established by Bennett et al. [2] on the number of incidences between points and flats:
Corollary 1.5 (Bennett et al. [2]). Let be a set of points, and a set of k-flats in
P F
Fd+k. Then the number of incidences between and satisfies
q P F
I( , ) |P||F| qdk/2 .
P F − qk ≤ |P||F|
(cid:12) (cid:12)
(cid:12) (cid:12) p
(cid:12) (cid:12)
It follows from Theorem(cid:12)1.3 and Theorem(cid:12)1.4 that if 2qk(d+2), then and
|P||V| ≥ P V
determine at least one incidence. Also if 2qk(d+2), then the number of incidences
|P||V| ≫
is close to the expected value /qk.
|P||V|
There are some applications of Corollary 1.5 in combinatorial geometry problems, for
instance, the number of congruent classes of triangles determined by a set of points in F2
q
in [2], and the number of right angles determined by a point set in Fd in [27].
q
3
When k = 1, varieties of the form W become hypersurfaces in Fd, so they can be
a1,...,ak q
written as
W = V h(x)+a xb1 + +a xbd +a , a = (a ,...,a ) Fd. (1.2)
a 1 1 ··· d d d+1 1 d ∈ q
(cid:16) (cid:17)
Therefore, we obtain the following bound on the number of incidences between points and
hypersurfaces:
Theorem 1.6. Let be a set of points in Fd, and a set of hypersurfaces of the form
P q S
W . Then the number of incidences between and satisfies
a
P S
I( , ) |P||S| qd/2 .
P S − q ≤ |P||S|
(cid:12) (cid:12)
(cid:12) (cid:12) p
(cid:12) (cid:12)
When h(x) = x21 + ··· +(cid:12)x2d, a = (1,...,1(cid:12)) and b1 = ... = bd = 2, as a consequence
of Theorem 1.6, we recover the bound on the number of incidences between points and
spheres obtained in [5, 26].
Corollary 1.7 (Cilleruelo et al. [5]). Let be a set of points, and a set of spheres
P S
with arbitrary radii in Fd. Then the number of incidences between and satisfies
q P S
I( , ) |P||S| qd/2 .
P S − q ≤ |P||S|
(cid:12) (cid:12)
(cid:12) (cid:12) p
(cid:12) (cid:12)
Theorem 1.7 also has various applications in several combinatorial problems over finite
(cid:12) (cid:12)
fields, for instance, Erd˝os distinct distance problem, the Beck type theorem for points and
circles, and subset without repeated distance, see [5, 26] for more details.
1.2 Pinned values and Distinct radii
Pinned values problem: The distance function between two points x and y in Fd,
q
denoted by x y , is defined as x y = (x y )2 + +(x y )2. Although it
1 1 d d
|| − || || − || − ··· −
is not a norm, the function x y has properties similar to the Euclidean norm (for
|| − ||
example, it is invariant under orthogonal matrices).
Bourgain, Katz, and Tao [3] were the first to consider the the finite analogue of the
classical Erd˝os distinct distance problem, namely to determine the smallest possible car-
dinality of the set ∆Fq(E) = {||x−y|| = (x1 −y1)2 + ···+ (xd −yd)2: x,y ∈ E} ⊂ Fq,
where Fd. More precisely, they proved that ∆F ( ) & 1/2+ǫ, where = qα and
E ⊂ q | q E | |E| |E|
ǫ > 0 is a small constant depending on α.
Iosevich and Rudnev [17] studied the following question: how large does Fd,
E ⊂ q
d 2 have to be, so that ∆F ( ) contains a positive proportion of the elements of
≥ q E
F . They proved that if Fd such that & Cqd/2 for sufficiently large C, then
q E ⊂ q |E|
∆F ( ) = Ω min q,q−(d−1)/2 (in other words, for any sufficiently large Fd, the
| q E | |E| E ⊆ q
set ∆Fq(E) co(cid:0)ntain(cid:8)s a positive pro(cid:9)p(cid:1)ortion of the elements of Fq). From this, one obtains
that that if & q(d+1)/2, then ∆F ( ) & q. This is in fact directly related to Falconer’s
|E| | q E |
4
result [8] in Euclidean space, saying that for every set with Hausdorff dimension greater
E
that (d+1)/2, the distance set is of positive measure.
Hartetal. [11]provedthattheexponent(d+1)/2isthebestpossibleinodddimensions,
although in even dimensions, it might still be place for improvement. Chapman et al. [6]
showedthatifaset F2 satisfies q4/3,then ∆F ( ) containsapositiveproportion
E ⊆ q |E| ≥ | q E |
of the elements of F . In the same paper it was also proved that for any set of points
q
P
in Fd with q(d+1)/2, there exists a subset ′ in , such that ′ = (1 o(1)) ,
q |P| ≥ P P |P | − |P|
and for any y ′, ∆F ( ,y) & q, where ∆F ( ,y) = x y : x . (which is
∈ P | q P | q P {|| − || ∈ P}
the pinned distance problem)
Let Q(x) be a non-degenerate quadratic form. For a fixed non-square element λ
∈
F 0 , the quadraic form Q(x) can be written as
q
\{ }
Q(x) = x2 x2 +x2 x2 + +x2 ǫx2 , if d = 2m,
1 − 2 3 − 4 ··· 2m−1 − 2m
and
Q(x) = x2 x2 +x2 x2 + +x2 x2 +ǫx2 , if d = 2m+1,
1 − 2 3 − 4 ··· 2m−1 − 2m 2m+1
where ǫ 1,λ , see [18] for more details.
∈ { }
Given a point q Fd and a set of points Fd, we define the pinned distance set
∈ q P ⊆ q
determined by Q(x) and q as ∆ ( ,q) = Q(p q): p . Using methods from
Q
P { − ∈ P}
spectral graph theory, the fifth listed author obtained the following:
Theorem 1.8 (Vinh [35]). Let be a set of points in Fd such that q(d+1)/2, then
P q |P| ≥
there exists a subset S such that S = (1 o(1)) , and for any y S, we have
⊂ P | | − |P| ∈
∆ ( ,y) & q.
Q
| P |
In our paper, as an application of Theorem 1.3 and using a similar approach to the
one in [5], we generalize Theorem 1.8 to non-degenerate polynomials. If F(x,y) is a
polynomial in F [x ,...,x ,y ,...,y ], we say that F(x,y) is non-degenerate if F(x,y)
q 1 d 1 d
can be written as
F(x,y) := g(x,y)+(xb1,...,xbd)M(yc1,...,ycd)T,
1 d 1 d
where g(x,y) = g (x)+g (y) F [x ,...,x ,y ,...,y ], M is a d d invertible matrix,
1 2 q 1 d 1 d
∈ ×
and gcd(c ,q 1) = 1 for all 1 i d.
i
− ≤ ≤
Theorem 1.9. Let F(x,y) be a non-degenerate polynomial and be a set of points in
P
Fd such that (√1 c2/c2)q(d+1)/2 for some constant 0 < c < 1. Then there is
q |P| ≥ −
′ such that ′ (1 c) , and for any y ′, ∆ ( ,y) (1 c)q, where
F
P ⊂ P |P | ≥ − |P| ∈ P | P | ≥ −
∆ ( ,q) = F(p,q): p .
F
P { ∈ P}
Corollary 1.10. Let F(x,y) be a non-degenerate polynomial and , be sets of points
P Q
in Fd such that 2√3qd+1 for some constant 0 < c < 1. Then there is ′
q |P||Q| ≥ P ⊂ P
such that ′ /2, and for any y ′, ∆ ( ,y) q/2, where ∆ ( ,q) =
F F
|P | ≥ |P| ∈ P | Q | ≥ Q
F(p,q): p .
{ ∈ Q}
5
The Beck type theorem for points and spheres: Let be a set of points in F2.
P q
Iosevich, Rudnev, and Zhai [19] made the first investigation on the finite fields analogue
of the Beck type theorem for points and lines in F2. More precisely, they proved that if
q
64qlogq, then the number of distinct lines determined by is at least q2/8. In [23],
|P| ≥ P
Lund and Saraf improved the condition of the cardinality of to 3q. Recently, Cilleruelo
P
et al. [5] studied the Beck type theorem for points and circles in F2 by employing the
q
lower bound on the number of incidences between points and circles in F2. Formally, their
q
result is as follows.
Theorem 1.11 (Cilleruelo et al. [5]). Let be a set of points in F2. If 5q, then
P q |P| ≥
the number of distinct circles determined by is at least 4q3/9.
P
As a consequence of Theorem 1.11, we obtain the following result.
Theorem 1.12. Let be a set of 5q points in F2. Then the number of distinct radii of
P q
circles determined by is at least 4q/9.
P
Note that it is hard to generalize Theorem 1.11 in higher dimensional cases by their
arguments. In the following theorem, we will give an approach to address this problem
by using a result on the number of pinned distinct distances.
Theorem 1.13. Let be a set of 8q2 points in F3. Then the number of distinct spheres
P q
determined by is at least q4/9.
P
As a consequence of Theorem 1.13, we obtain the following result on the number of
distinct radii of spheres determined by a set of points in F3.
q
Theorem 1.14. Let be a set of 8q2 points in F3. Then the number of distinct radii of
P q
spheres determined by is at least q/9.
P
Remark 1.15. We note that one can follow the proof of Theorem 1.13 to prove that there
exist constants c = c(d) and c′ = c′(d) such that there are at least cqd+1 d-dimensional
spheres determined by a set of c′qd−1 points in Fd.
q
1.3 Distinct distances between points and lines
As already mentioned in the abstract, we use the same approach to address the finite
field variants of two recent results due to Sharir et al. [29], involving distances between
points and lines. The first bound is a lower bound for the minimum number of distinct
distances between a set of points and a set of lines, both in the plane. A second result is a
lower bound for the minimum number of distinct distances between a set of non-collinear
points and the lines that they span.
Theorem 1.16 (Sharir et al. [29]). For m1/2 n m2, the minimum number D(m,n)
≤ ≤
of point-line distances between m points and n lines in R2 satisfies D(m,n) = Ω(m1/5n3/5)
Theorem 1.17 (Sharir et al. [29]). The minimum number H(m) of point-line distances
between m non-collinear points and their spanned lines satisfies H(m) = Ω(m4/3).
6
Intheplaneover finitefields, alineax+by+c = 0isdegenerate ifandonlyifa2+b2 = 0.
Similarly, a hyperplane a x + + a x + a = 0 in Fd is degenerate if and only if
1 1 ··· d d d+1 q
a2+ +a2 = 0. Forapoint p = (x ,y ) F2 anda non-degenerateline l : ax+by+c = 0
1 ··· d p p ∈ q
in F2, let d(p,l) denote the distance function between p and l, defined as
q
(ax +by +c)2
p p
d(p,l) = .
a2 +b2
For a set of points in F2 and a line l, set ∆F ( ,l) = d(p,l) : p . Distances
P q q P { ∈ P}
between points and non-degenerate lines are preserved under rotations and translations.
Similarly, for a point p = (x1,x2,...,xd) Fd and a non-dengenerate hyperplane
p p p ∈ q
h : a x + +a x +a = 0, we define the point-hyperplane distance
1 1 d d d+1
···
d(p,h) = (a x1 + +a xd +a )2/(a2 + +a2).
1 p ··· d p d+1 1 ··· d
For a set of points P in Fdq and a hyperplane h, we let ∆Fq(P,h) = {d(p,h) : p ∈ P}.
We prove that under a similar condition as in the result due to Chapman et al. [6] on
the number of distinct distances between points in F2, the set of distances between and
q P
contains a positive proportion of the elements of F .
q
L
Theorem 1.18. Let be a set of points and be a set of non-degenerate lines in F2,
P L q
such that
4(1 c2)
− q8/3
|P||L| ≥ (1/2 (1 c2))2
− −
with 1 c2 < 1/4. Then there exists a subset ′ of with ′ = (1 o(1)) , so that
− L L |L| − |L|
∆F ( ,l) & q, for each line l in ′.
| q P | L
Combining a finite field variant of Beck’s theorem (which can be found in [23]) with
Theorem 1.18, we obtain the following bound on the number of distinct distances between
a set of points and their spanned lines.
Corollary 1.19. Let be a set of points in F2 with 3q, and let be the set of
P q |P| ≥ L
lines spanned by in F2. Then there exists a subset ′ of with ′ = (1 o(1)) , so
P q L L |L| − |L|
that ∆F ( ,l) & q, for each line l in ′.
| q P | L
By similar arguments as in the proof of Theorem 1.18, we obtain a similar result on
the number of distinct distances between points and hyperplanes in d-dimensional vector
space over finite fields as follows.
Theorem 1.20. Let be a set of points in Fd, and be a set of non-degenerate hyper-
P q H
planes in Fd, such that
q
4(1 c2)
− q4d/3,
|P||H| ≥ (1/2 (1 c2))2
− −
with 1 c2 < 1/4. Then there exists a subset ′ of with ′ = (1 o(1)) , so that
− H H |H| − |H|
∆F ( ,h) & q, for each line h in ′.
| q P | H
7
2 Tools
This section contains a couple of notions and theorems that we use as tools in the proofs
of our main results. We fist state the well-known Schwartz-Zippel Lemma (for proof refer
to Theorem 6.13 in [24]).
Lemma 2.1 (Schwartz-Zippel). Let P(x) be a non-zero polynomial of degree k. Then
x Fd : P(x) = 0 kqd−1.
{ ∈ q } ≤
We say that a bipartite gr(cid:12)aph is biregular if in(cid:12)both of its two parts, all vertices have
(cid:12) (cid:12)
the same degree. If A is one of the two parts of a bipartite graph, we write deg(A) for
the common degree of the vertices in A. Label the eigenvalues so that λ λ
1 2
| | ≥ | | ≥ ··· ≥
λ . Note that in a bipartite graph, we have λ = λ . The following variant of the
n 2 1
| | −
expander mixing lemma is proved in [7]. We include the proof of this result for the sake
of completeness of the paper.
Lemma 2.2 (Expander mixing lemma). Let G be a bipartite graph with parts A,B
such that the vertices in A all have degree a and the vertices in B all have degree b. Then,
for any two sets X A and Y B, the number of edges between X and Y, denoted by
⊂ ⊂
e(X,Y), satisfies
a
e(X,Y) X Y λ X Y ,
3
− B | || | ≤ | || |
(cid:12) | | (cid:12)
where λ is the third eigen(cid:12)value of G. (cid:12) p
3 (cid:12) (cid:12)
(cid:12) (cid:12)
Proof. We assume that the vertices of G are labeled from 1 to A + B , and we denote
| | | |
by M the adjacency matrix of G having the form
0 N
M = ,
Nt 0
(cid:20) (cid:21)
where N is the A B 0 1 matrix, with N = 1 if and only if there is an edge between
ij
| |×| | −
i and j. First, let us recall some properties of the eigenvalues of the matrix M. Since
all vertices in A have degree a and all vertices in B have degree b, all eigenvalues of M
are bounded by √ab. Indeed, let us denote the L vector norm by , and let e be
1 1 v
||· ||
the unit vector having 1 in the position corresponding to vertex v and zeroes elsewhere.
One can observe that M2 e ab, so the absolute value of each eigenvalue of M
v 1
|| · || ≤
is bounded by √ab. Let 1 denote the column vector of size A + B having 1s in the
X
| | | |
positions corresponding to the set of vertices X and 0s elsewhere. Then, we have that
M(√a1 +√b1 ) = b√a1 +a√b1 = √ab(√a1 +√b1 ),
A B B A A B
M(√a1 √b1 ) = b√a1 a√b1 = √ab(√a1 √b1 ),
A B B A A B
− − − −
which implies that λ = √ab and λ = √ab are the first and second eigenvalues,
1 2
−
corresponding to the eigenvectors (√a1 +√b1 ) and (√a1 √b1 ).
A B A B
−
Let W⊥ be a subspace spanned by the vectors 1 and 1 . Since M is a symmetric ma-
A B
trix, the eigenvectors of M, except √a1 +√b1 and √a1 √b1 , span W. Therefore,
A B A B
−
for any u W, Mu W, and Mu λ u . Let us now remark the following facts:
3
∈ ∈ || || ≤ || ||
8
0 J
1. Let K be a matrix of the form , where J is the A B all-ones matrix. If
J 0 | |×| |
(cid:20) (cid:21)
u W, then Ku = 0 since every row of K is either 1T or 1T.
∈ A B
2. If w W⊥, then (M (a/ B )K)w = 0. Indeed, it follows from the facts that
∈ − | |
a A = b B , and M1 = b1 = (a/ B )K1 , M1 = a1 = (a/ B )K1 .
A B A B A B
| | | | | | | |
Since e(X,Y) = 1TM1 and X Y = 1TK1 ,
Y X | || | Y X
a a
e(X,Y) X Y = 1T(M K)1 .
− B | || | Y − B X
(cid:12) | | (cid:12) (cid:12) | | (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
For any vector v, let v¯(cid:12) be the orthogonal p(cid:12)roje(cid:12)ction onto W, so th(cid:12)at v W, and v v
(cid:12) (cid:12) (cid:12) (cid:12) ∈ − ∈
W⊥. Thus
a a
1T(M K)1 = 1T(M K)1 = 1TM1 = 1 TM1 , so
Y − B X Y − B X Y X Y X
| | | |
a
e(X,Y) X Y λ 1 1 .
3 X Y
− B | || | ≤ || || || ||
(cid:12) | | (cid:12)
(cid:12) (cid:12)
Since
(cid:12) (cid:12)
1 = 1 (cid:12) ((1 1 )/(1 1 (cid:12)))1 = 1 ( X / A )1 ,
X X X A A A A X A
− · · − | | | |
we have 1 = X (1 X / A ). Similarly, 1 = Y (1 Y / B ).
X Y
|| || | | −| | | | || || | | −| | | |
In other words,
p p
a
e(X,Y) X Y λ X Y (1 X / A )(1 Y / B ),
3
− B | || | ≤ | || | −| | | | −| | | |
(cid:12) | | (cid:12)
(cid:12) (cid:12) p
which comple(cid:12)tes the proof of the l(cid:12)emma.
(cid:12) (cid:12)
3 Proofs of Theorems 1.3, 1.4, and Corollary 1.7
We start by proving the following lemma.
Lemma 3.1. Foranytwo k-tuples ofvectors (a ,...,a ) = (c ,...,c ), wehaveV =
1 k 6 1 k a1,...,ak 6
V .
c1,...,ck
Proof. Since (a ,...,a ) = (c ,...,c ), without loss of generality, we can assume that
1 k 1 k
6
a = c . Therefore,
1 1
6
f (x,a ) f (x,c ) = (a c )xb11 + +(a c )xb1d +a c ,
1 1 − 1 1 11 − 11 1 ··· 1d − 1d d 1(d+1) − 1(d+1)
is a non-zero polynomial of degree at most q 1 in F [x ,...,x ].
q 1 d
−
By Lemma 2.1, the cardinality of V(f (x,a ) f (x,c )) is at most (q 1)qd−1 < qd.
1 1 1 1
− −
Let us observe that if V V , then V(f (x,a ) f (x,c )) = qd. This is
a1,...,ak ≡ c1,...,ck | 1 1 − 1 1 |
indeed the case since each variety contains exactly qd points in Fd Fk. Thus, we obtain
q × q
V (x f (x,a ),...,x f (x,a )) = V (x f (x,c ),...,x f(x,c )),
d+1 1 1 d+k k k d+1 1 1 d+k k
− − 6 − −
which completes the proof of the lemma.
9
We define the bipartite graph G = (A B,E) as follows. The first vertex part A
∪
is (F )d (F )k and the second vertex part B is the set of all varieties V with
q × q a1,...,ak
(a ,...,a ) Fd+1 k. We draw an edge between a point p A and a variety v B
1 k ∈ q ∈ ∈
if and only if p v. It is easy to check that G is biregular with deg(A) = qdk and
(cid:0) ∈ (cid:1)
deg(B) = qd.
Lemma 3.2. Let λ be the third eigenvalue of the adjacency matrix of G. Then λ
3 3
| | ≤
qdk/2.
0 N
Proof. Let M betheadjacencymatrixofG, so M = , where N isaqd+k q(d+1)k
NT 0 ×
(cid:20) (cid:21)
matrix, with N = 1 if p v, N = 0 if p v.
pv pv
∈ 6∈
0 J
Let J be the qd+k qk(d+1) all-one matrix and K = . We prove that M satisfies
× JT 0
(cid:20) (cid:21)
M3 = qdkM +(qd 1)qk(d−1)K.
−
If v is an eigenvector corresponding to the third eigenvalue λ , then Kv = 0. Therefore
3
from the equation above one obtains that λ3 = qdkλ , which implies that λ = qdk.
3 3 | 3|
Let us observe that the (p,v)-entry of M3 equals the number of walks of length three
p
from p A to v B, that is the number of quadruples (p,v′,p′,v), where p,p′
∈ ∈ ∈
A,v,v′ B, and (p,v′),(p′,v′),(p′,v) are edges of G.
∈
Given two points p = (p ,...,p ) and p′ = (p′,...,p′ ), the varieties containing
1 d+k 1 d+k
both p and p′, and corresponding to k-tuples (a ,...,a ) F(d+1)k satisfies
1 k q
∈
p = h (p ,...,p )+a pbi1 + +a pbid +a ,
d+i i 1 d i1 1 ··· id d id+1 (3.1)
p′ = h (p′,...,p′)+a (p′)bi1 + +a (p′)bid +a ,
d+i i 1 d i1 1 ··· id d id+1
for all 1 i k. Thus, for each 1 i k, we have
≤ ≤ ≤ ≤
p p′ = h (p ,...,p ) h (p′,...,p′)+a (pbi1 (p′)bi1)+ +a (pbid (p′)bid). (3.2)
d+i− d+i i 1 d − i 1 d i1 1 − 1 ··· id d − d
Let us observe that for each a F , if gcd(r,q 1) = 1, then the equation xr = ar has
q
∈ −
the unique solution x = a. Thus if p = p′ for all 1 i d, then there exists at least one
i i ≤ ≤
variety containing both p and p′ if and only if p = p′ for all 1 i k. This implies
d+i d+i ≤ ≤
that p = p′.
We now count the number of walks of length three as follows. If p v, then we can
6∈
choose p′ = p in v such that p = p′ for some 1 i d (otherwise, there is no v′
6 i 6 i ≤ ≤
containing both p and p′). We assume that p = p′, so pbi1 = (p′)bi1. Therefore, for each
1 6 1 1 6 1
choice of (a ,...,a ), a is determined uniquely by (3.2), and a is determined by
i2 id i1 id+1
any equation in (3.1). In this case, the number of walks of length three is (qd 1)qk(d−1).
−
If p v, then again there are (qd 1)qk(d−1) walks with p = p′. Now we can choose
∈ − 6
p = p′. In this case, the number of walks equals the degree of p. Thus if p v, then the
∈
number of walks of length three from p to v is (qd 1)qk(d−1) +qdk.
−
In conclusion, M satisfies M3 = qdkM +(qd 1)qk(d−1)K, which completes the proof of
−
the lemma.
10