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An elementary bound for the number of points of a hypersurface over a finite field
Finite Fields and Their Applications 20 (2013) 76–83
Contents lists available at SciVerse ScienceDirect
Finite Fields and Their Applications www.elsevier.com/locate/ffa
An elementary bound for the number of points of a hypersurface over a finite field Masaaki Homma a,∗,1 , Seon Jeong Kim b,2 a b
Department of Mathematics and Physics, Kanagawa University, Hiratsuka 259-1293, Japan Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 30 July 2012 Accepted 14 November 2012 Available online 24 November 2012 Communicated by Gary L. Mullen MSC: 14J70 11G25 05B25
a b s t r a c t We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve. Our bound is the best one for irreducible hypersurfaces that is linear on their degrees, because, for each finite field, there are at least two irreducible hypersurfaces of different degrees that reach our bound. © 2012 Elsevier Inc. All rights reserved.
Keywords: Hypersurface Finite field Number of points
1. Introduction In the series of papers [4–6], we settled the Sziklai conjecture [11]. Properly speaking, for any plane curve C of degree d over the finite field Fq of q elements without Fq -linear components, the number N q (C ) of Fq -points of C is bounded by
N q (C ) (d − 1)q + 1 except for a curve of degree 4 over F4 . Moreover some curves actually achieve this bound.
*
Corresponding author. E-mail addresses: [email protected] (M. Homma), [email protected] (S.J. Kim). 1 Partially supported by Grant-in-Aid for Scientific Research (24540056), JSPS. 2 Partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0027246). 1071-5797/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ffa.2012.11.002
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
77
We want to establish an analogue of this fact for a hypersurface in Pn without an Fq -linear component. The bound (1) below is also valid for n = 2, but not tight; in other words, it is interesting only when n 3. Notation 1.1. The number of points of the r-dimensional finite projective space Pr (Fq ) over Fq is qr + qr −1 + · · · + 1 = (qr +1 − 1)/(q − 1), which is denoted by θq (r ). We understand θq (−1) = 0. Theorem 1.2. Let X ⊂ Pn be a hypersurface of degree d defined over Fq without an Fq -linear component. Then
N q ( X ) (d − 1)qn−1 + dqn−2 + θq (n − 3).
(1)
Moreover if n 3, equality holds for a couple of irreducible hypersurfaces. In earlier literatures, we can find several bounds for N q ( X ). In the last section, we compare our bound with those known bounds, namely the Weil conjecture [13] established by Deligne [2], a bound due to Segre [8], Serre [9] and Sørensen [10], and its generalization obtained by Thas [12]. 2. Set-theoretic degrees In [3], we introduced a pure combinatorial notion called the s-degree, which is an abbreviation for the set-theoretic degree. We generalize this notion slightly. In this section, we consider subsets of the n-dimensional finite projective space Pn (Fq ) over Fq . Definition 2.1. For a subset S ⊂ Pn (Fq ), the i-th s-degree of S is defined by
#
di ( S ) = max
( S ∩ M i ) M i is an Fq -linear subspace of codimension i .
The i-th s-degree makes sense only when 1 i n. In [3], we showed the following fact. Proposition 2.2. Let d1 = d1 ( S ). Then the number N of points of S is bounded by
N (d1 − 1)q + 1 +
d1 − 1
θq (n − 2)
(2)
.
The next bound involves the second s-degree d2 . Proposition 2.3. Let S ⊂ Pn (Fq ), d1 = d1 ( S ), and d2 = d2 ( S ). Then the number N of points of S is bounded by
N (d1 − d2 )q + d1 . Proof. Let L 0 be a linear subspace of codimension 2 such that hyperplanes of Pn (Fq ) containing L 0 . Then
N=
#
(3) #
( S ∩ L 0 ) = d2 , and Lˇ 0 the set of
( H ∩ S − d2 ) + d2
H ∈ Lˇ 0
(d1 − d2 )(q + 1) + d2 = (d1 − d2 )q + d1 . This completes the proof.
2
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M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
3. Hypersurfaces in P n This section consists of two parts. In the first part, we prove the inequality (1) in Theorem 1.2, and in the second part, we exhibit examples that attain this bound. 3.1. Proof of (1) If d q + 1, then
(d − 1)qn−1 + dqn−2 + θq (n − 3) qn + (q + 1)qn−2 + θq (n − 3) = θq (n). Hence the inequality in Theorem 1.2 is trivial when d q + 1. So we may assume that d q + 1. We prove the inequality by induction on n. When n = 2, the inequality means that N q ( X ) (d − 1)q + d, which is obviously true (see, for example, [4]). For n 3, we divide the situation into two cases. (Case 1) Suppose that there is an Fq -linear subspace L 0 of codimension 2 in Pn which is contained in X . Therefore the second s-degree of X (Fq ) is θq (n − 2). For any hyperplane H of Pn over Fq , X ∩ H is a hypersurface over Fq of degree d in H Pn−1 because X has no Fq -linear component. By the theorem of Segre–Serre–Sørensen (see Section 4 below),
N q ( X ∩ H ) dqn−2 + θq (n − 3), which means that the first s-degree of X (Fq ) is at most dqn−2 + θq (n − 3). Hence, by (3),
N q ( X ) dqn−2 + θq (n − 3) − θq (n − 2) q + dqn−2 + θq (n − 3)
= dqn−2 − qn−2 q + dqn−2 + θq (n − 3) = (d − 1)qn−1 + dqn−2 + θq (n − 3).
(Case 2) Suppose that there is no Fq -linear subspace of codimension 2 in Pn which is contained in X . Then for any hyperplane H of Pn over Fq , X ∩ H is a hypersurface of degree d in H Pn−1 without an Fq -linear component. Hence
N q ( X ∩ H ) (d − 1)qn−2 + dqn−3 + θq (n − 4) by the induction hypothesis. Therefore, by (2),
N q ( X ) (d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 q + 1
+
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 . θq (n − 2)
Since
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 = (d − 1) qn−2 + qn−3 + θq (n − 4) + qn−3 − (d − 2)θq (n − 4) − 1 < (d − 1)θq (n − 2) + qn−3 ,
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
79
we have
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 qn−3 (d − 1) + = d − 1. θq (n − 2) θq (n − 2)
Therefore
N q ( X ) (d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 q + 1 + (d − 1)
= (d − 1)qn−1 + dqn−2 + θq (n − 4) − 1 q + d
(d − 1)qn−1 + dqn−2 + qθq (n − 4) + 1 (because d q + 1) = (d − 1)qn−1 + dqn−2 + θq (n − 3). This completes the proof.
2
3.2. Example Our bound (1) is loose for n = 2, but is tight for n 3 even if we consider only irreducible hypersurfaces, as we see below. Lemma 3.1. Let X be a hypersurface of degree d in Pn over Fq . Regard this Pn as a hyperplane in Pn+1 defined by a linear equation over Fq , and take an Fq -point P 0 ∈ Pn+1 \ Pn . Then the cone X˜ over X with vertex P 0 is also a hypersurface of degree d in Pn+1 over Fq , and N q ( X˜ ) = qN q ( X ) + 1. Proof. If we choose coordinates X 0 , . . . , X n+1 of Pn+1 so that the hyperplane Pn is defined by X n+1 = 0 and P 0 = (0, . . . , 0, 1), the equation f ( X 0 , . . . , X n ) = 0 defining X in Pn also defines X˜ in Pn+1 . Since
X˜ = s(x0 , . . . , xn , 0) + t (0, . . . , 0, 1) f (x0 , . . . , xn ) = 0, (s, t ) ∈ P1 , we have N q ( X˜ ) = qN q ( X ) + 1.
2
Corollary 3.2. Under the assumption in Lemma 3.1, if N q ( X ) achieves the upper bound (1) in Pn , then so does N q ( X˜ ) in Pn+1 . Proof. Actually we have
N q ( X˜ ) = q (d − 1)qn−1 + dqn−2 + θq (n − 3) + 1
= (d − 1)qn + dqn−1 + qθq (n − 3) + 1 = (d − 1)qn + dqn−1 + θq (n − 2) by Lemma 3.1.
2
By this corollary, once an irreducible hypersurface over Fq in Pn achieves the upper bound (1), there exists one of the same degree in Pn+1 attaining the upper bound. So, to confirm the additional statement of Theorem 1.2, it is enough to consider surfaces in P3 .
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M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
Example 3.3. Let Q be the image of the Segre embedding of P1 × P1 in to P3 over Fq , which is a hyperbolic quadric surface. In other words, under suitable choice of coordinates X 0 , . . . , X 3 of P3 , Q is defined by X 0 X 1 − X 2 X 3 = 0. Obviously, N q ( Q ) = (q + 1)2 = (2 − 1)q2 + 2q + θq (0). √
Example 3.4. Suppose q is a square number. It is known that the Hermitian surface H : X 0
q+1
+ √ 3 2 X1 + X2 + X3 = 0 in P has (q + 1)(q q + 1) Fq -points [1]. Since ( q + 1 − 1)q + √ √ ( q + 1)q + θq (0) = (q + 1)(q q + 1), N q ( H ) attains the upper bound (1). √
√
q+1
√
q+1
√
q+1
Example 3.5. Let F be a space filling surface q
q
q
q
X 0 X 1 − X 0 X 1 + X 2 X 3 − X 2 X 3 = 0. Then N q ( F ) = θq (3) = (q + 1 − 1)q2 + (q + 1)q + θq (0). Remark 3.6. Surfaces which appeared in the above Examples are nonsingular and in P3 , hence they are irreducible. 4. Comparison with other bounds 4.1. A bound from the Weil conjecture The Weil conjecture [13] established by Deligne [2] provides a bound for the number of Fq -points of nonsingular varieties defined over Fq , involving their Betti numbers. The Betti numbers b0 , b1 , . . . , b2(n−1) of a nonsingular hypersurface of degree d in Pn are the following [7, p. 163]:
• For i with 0 i < n − 1, bi = b2(n−1)−i = • Let b =
1 if i is even, 0 if i is odd;
bn−1 − 1 if n − 1 is even, 1 Then b = d− ((d − 1)n − (−1)n ). d bn−1 if n − 1 is odd.
Hence the Weil conjecture implies the following bound [2, Theorem 8.1]. Theorem 4.1 (Deligne). Let X be a nonsingular hypersurface of degree d in Pn defined over Fq . Then
N q ( X ) θq (n − 1) + b q
n −1 2
(4)
.
Now we compare our bound (1) with (4). Proposition 4.2. Suppose n 3. If d
√
q + 2, then
(d − 1)qn−1 + dqn−2 + θq (n − 3) < θq (n − 1) + b q
n −1 2
.
Proof. Let
F (x) =
=
x − 1 x
n −1 (x − 1)n − (−1)n q 2 + θq (n − 1) − (x − 1)qn−1 + xqn−2 + θq (n − 3)
x − 1 x
(x − 1)n − (−1)n − xq
n −1 2
− xq
n −3 2
q
n −1 2
+ qn−1 .
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
We want to prove that F (x) > 0 if x
√
q + 2. When x =
81
√
q + 2,
√
√ √ √ n −1 √ √ n −3 √ n −1 q + 1 √ n −1 √ ( q + 1)n − (−1)n − ( q + 2) q F ( q + 2) > √ q − ( q + 2) q + q q+2
√ √ n −2 √ n −3 √ n −1 q + 1 √ n −1 n n −1 −2 q + q + q =√ (n − 1) q q+2 2 3 n −4 n √ k n + q + 1 − (−1) k =1
k
√ √ n −3 q + 1 √ n −1 n n + −4 √ q n+ q > 0. q+2 2 3 Let F (x) be the derivative of F (x) by x. If x
F (x) =
(x − 1)n (nx + 1) − (−1)n x2
n −1
> ( x − 1)
−1−
√
n −1
q
−
√
−
√
q
√
q + 2, then
n −1
−
√
n −3
q
√
n −1
q
n−3 √ n−1
q
q
because
(x − 1)(nx + 1) x2
√ √ n −1 √ n −3 > 0. − q ( q + 1)n−1 − 1 − q Hence, for any x
√
q + 2, we have F (x) > 0.
>1
2
The following corollary was already remarked by Sørensen [10]. Corollary 4.3. If the number of Fq -points of a nonsingular hypersurface of degree d meets the bound (4) from √ the Weil conjecture, then d q + 1.
√
Proof. When n = 2, the bound (4) is N q ( X ) q + 1 + (d − 1)(d − 2) q, and the Sziklai bound says that N q ( X ) (d − 1)q + 1. Since
√
√
√
q + 1 + (d − 1)(d − 2) q − (d − 1)q + 1 = d − ( q + 1) (d − 2) q, any irreducible plane curve of degree d > statement is just the above proposition. 2
√
q + 1 cannot attain the bound (4). When n 3, this
4.2. Segre–Serre–Sørensen bound Without any restrictions on a hypersurface over Fq , the best bound was obtained by Serre [9], which is a generalization of Segre’s old result [8] for plane curves. Sørensen [10] also proved the same inequality as Serre’s.3 Theorem 4.4 (Segre–Serre–Sørensen). Let X ⊂ Pn be a hypersurface of degree d defined over Fq . Then N q ( X ) dqn−1 + θq (n − 2). Moreover, if d q, equality holds if and only if X is a union of d hyperplanes over Fq that contain a common linear subspace of codimension 2 in Pn .
3
The date on Serre’s letter to Tsfasman is 24 July 1989, and the received date of Sørensen’s paper is 17 January 1990.
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M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
Remark 4.5. The assumption d q in the additional statement of the Segre–Serre–Sørensen theorem can be weakened as d q + 1 if n = 2 [8, II, §6, Observation IV], but cannot be when n 3 as Serre noted [9, Remark 2]. In the literature cited above, Serre also remarked that if a hypersurface X of degree d does not split into d Fq -hyperplanes, then a stronger bound than one in Theorem 4.4 holds:
N q ( X ) dqn−1 + θq (n − 2) − (q + 1 − d).
(5)
Our bound (1) is tighter than (5). In fact,
dqn−1 + θq (n − 2) − (q + 1 − d) − (d − 1)qn−1 + dqn−2 + θq (n − 3)
= (q + 1 − d) qn−2 − 1 .
4.3. Bounds of K. Thas In [12], Thas made refinements on Theorem 4.4. Those refinements involve a new invariant k X for a hypersurface X ⊂ Pn of degree d defined over Fq . The invariant k X is the maximal dimension of an Fq -linear subspace which is contained in X . He took two ways of finding an upper bound for N q ( X ); the first bound goes up according as k X increases, and the second goes down. Finally, he obtained a bound by mixing these two methods, whose assumption fits in ours. Theorem 4.6 (Thas). Let X be a hypersurface in Pn of degree d defined over Fq , and k = k X . If 0 < k n − 2, then
N q ( X ) dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1 . Moreover, if 0 < k < n − 2, strict inequality holds in (6). Remark 4.7. For the k = 0 case, he only showed a weak bound (5). Now we compare Thas’s bound (6) with ours (1). Proposition 4.8. For n 3, k n − 2 and d < q + 1, we have
(d − 1)qn−1 + dqn−2 + θq (n − 3) dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1 , and equality holds if and only if k = 0 or k = n − 2. It is easy to see that
dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1
= (d − 1)qn−1 + dqn−2 + θq (n − 3) + (q + 1 − d) qk − 1 qn−2−k − 1 .
This completes the proof.
2
(6)
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
83
References [1] R.C. Bose, I.M. Chakravarti, Hermitian varieties in a finite projective space P G ( N , q2 ), Canad. J. Math. 18 (1966) 1161–1182. [2] P. Deligne, La conjecture de Weil, I, Publ. Math. Inst. Hautes Études Sci. 43 (1974) 273–307. [3] M. Homma, A bound on the number of points of a curve in a projective space over a finite field, in: M. Lavrauw, G.L. Mullen, S. Nikova, D. Panario, L. Storme (Eds.), Theory and Applications of Finite Fields, in: Contemp. Math., vol. 579, Amer. Math. Soc., Providence, 2012, pp. 103–110. [4] M. Homma, S.J. Kim, Around Sziklai’s conjecture on the number of points of a plane curve over a finite field, Finite Fields Appl. 15 (2009) 468–474. [5] M. Homma, S.J. Kim, Sziklai’s conjecture on the number of points of a plane curve over a finite field, II, in: G. McGuire, G.L. Mullen, D. Panario, I.E. Shparlinski (Eds.), Finite Fields: Theory and Applications, in: Contemp. Math., vol. 518, Amer. Math. Soc., Providence, 2010, pp. 225–234. [6] M. Homma, S.J. Kim, Sziklai’s conjecture on the number of points of a plane curve over a finite field, III, Finite Fields Appl. 16 (2010) 315–319. [7] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, second edition, Grad. Texts in Math., vol. 84, Springer-Verlag, New York, 1990. [8] B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959) 1–96. [9] J.-P. Serre, Lettre á M. Tsfasman, in: Journées Arithmétiques, Luminy, 1989, Astérisque 198–200 (1991) 351–353. [10] A.B. Sørensen, On the number of rational points on codimension-1 algebraic sets in Pn (Fq ), Discrete Math. 135 (1994) 321–334. [11] P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008) 41–43. [12] K. Thas, On the number of points of a hypersurface in finite projective space, after J.-P. Serre, Ars Combin. 94 (2010) 183–190. [13] A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949) 497–508.
Contents lists available at SciVerse ScienceDirect
Finite Fields and Their Applications www.elsevier.com/locate/ffa
An elementary bound for the number of points of a hypersurface over a finite field Masaaki Homma a,∗,1 , Seon Jeong Kim b,2 a b
Department of Mathematics and Physics, Kanagawa University, Hiratsuka 259-1293, Japan Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 30 July 2012 Accepted 14 November 2012 Available online 24 November 2012 Communicated by Gary L. Mullen MSC: 14J70 11G25 05B25
a b s t r a c t We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve. Our bound is the best one for irreducible hypersurfaces that is linear on their degrees, because, for each finite field, there are at least two irreducible hypersurfaces of different degrees that reach our bound. © 2012 Elsevier Inc. All rights reserved.
Keywords: Hypersurface Finite field Number of points
1. Introduction In the series of papers [4–6], we settled the Sziklai conjecture [11]. Properly speaking, for any plane curve C of degree d over the finite field Fq of q elements without Fq -linear components, the number N q (C ) of Fq -points of C is bounded by
N q (C ) (d − 1)q + 1 except for a curve of degree 4 over F4 . Moreover some curves actually achieve this bound.
*
Corresponding author. E-mail addresses: [email protected] (M. Homma), [email protected] (S.J. Kim). 1 Partially supported by Grant-in-Aid for Scientific Research (24540056), JSPS. 2 Partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0027246). 1071-5797/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ffa.2012.11.002
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
77
We want to establish an analogue of this fact for a hypersurface in Pn without an Fq -linear component. The bound (1) below is also valid for n = 2, but not tight; in other words, it is interesting only when n 3. Notation 1.1. The number of points of the r-dimensional finite projective space Pr (Fq ) over Fq is qr + qr −1 + · · · + 1 = (qr +1 − 1)/(q − 1), which is denoted by θq (r ). We understand θq (−1) = 0. Theorem 1.2. Let X ⊂ Pn be a hypersurface of degree d defined over Fq without an Fq -linear component. Then
N q ( X ) (d − 1)qn−1 + dqn−2 + θq (n − 3).
(1)
Moreover if n 3, equality holds for a couple of irreducible hypersurfaces. In earlier literatures, we can find several bounds for N q ( X ). In the last section, we compare our bound with those known bounds, namely the Weil conjecture [13] established by Deligne [2], a bound due to Segre [8], Serre [9] and Sørensen [10], and its generalization obtained by Thas [12]. 2. Set-theoretic degrees In [3], we introduced a pure combinatorial notion called the s-degree, which is an abbreviation for the set-theoretic degree. We generalize this notion slightly. In this section, we consider subsets of the n-dimensional finite projective space Pn (Fq ) over Fq . Definition 2.1. For a subset S ⊂ Pn (Fq ), the i-th s-degree of S is defined by
#
di ( S ) = max
( S ∩ M i ) M i is an Fq -linear subspace of codimension i .
The i-th s-degree makes sense only when 1 i n. In [3], we showed the following fact. Proposition 2.2. Let d1 = d1 ( S ). Then the number N of points of S is bounded by
N (d1 − 1)q + 1 +
d1 − 1
θq (n − 2)
(2)
.
The next bound involves the second s-degree d2 . Proposition 2.3. Let S ⊂ Pn (Fq ), d1 = d1 ( S ), and d2 = d2 ( S ). Then the number N of points of S is bounded by
N (d1 − d2 )q + d1 . Proof. Let L 0 be a linear subspace of codimension 2 such that hyperplanes of Pn (Fq ) containing L 0 . Then
N=
#
(3) #
( S ∩ L 0 ) = d2 , and Lˇ 0 the set of
( H ∩ S − d2 ) + d2
H ∈ Lˇ 0
(d1 − d2 )(q + 1) + d2 = (d1 − d2 )q + d1 . This completes the proof.
2
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3. Hypersurfaces in P n This section consists of two parts. In the first part, we prove the inequality (1) in Theorem 1.2, and in the second part, we exhibit examples that attain this bound. 3.1. Proof of (1) If d q + 1, then
(d − 1)qn−1 + dqn−2 + θq (n − 3) qn + (q + 1)qn−2 + θq (n − 3) = θq (n). Hence the inequality in Theorem 1.2 is trivial when d q + 1. So we may assume that d q + 1. We prove the inequality by induction on n. When n = 2, the inequality means that N q ( X ) (d − 1)q + d, which is obviously true (see, for example, [4]). For n 3, we divide the situation into two cases. (Case 1) Suppose that there is an Fq -linear subspace L 0 of codimension 2 in Pn which is contained in X . Therefore the second s-degree of X (Fq ) is θq (n − 2). For any hyperplane H of Pn over Fq , X ∩ H is a hypersurface over Fq of degree d in H Pn−1 because X has no Fq -linear component. By the theorem of Segre–Serre–Sørensen (see Section 4 below),
N q ( X ∩ H ) dqn−2 + θq (n − 3), which means that the first s-degree of X (Fq ) is at most dqn−2 + θq (n − 3). Hence, by (3),
N q ( X ) dqn−2 + θq (n − 3) − θq (n − 2) q + dqn−2 + θq (n − 3)
= dqn−2 − qn−2 q + dqn−2 + θq (n − 3) = (d − 1)qn−1 + dqn−2 + θq (n − 3).
(Case 2) Suppose that there is no Fq -linear subspace of codimension 2 in Pn which is contained in X . Then for any hyperplane H of Pn over Fq , X ∩ H is a hypersurface of degree d in H Pn−1 without an Fq -linear component. Hence
N q ( X ∩ H ) (d − 1)qn−2 + dqn−3 + θq (n − 4) by the induction hypothesis. Therefore, by (2),
N q ( X ) (d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 q + 1
+
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 . θq (n − 2)
Since
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 = (d − 1) qn−2 + qn−3 + θq (n − 4) + qn−3 − (d − 2)θq (n − 4) − 1 < (d − 1)θq (n − 2) + qn−3 ,
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
79
we have
(d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 qn−3 (d − 1) + = d − 1. θq (n − 2) θq (n − 2)
Therefore
N q ( X ) (d − 1)qn−2 + dqn−3 + θq (n − 4) − 1 q + 1 + (d − 1)
= (d − 1)qn−1 + dqn−2 + θq (n − 4) − 1 q + d
(d − 1)qn−1 + dqn−2 + qθq (n − 4) + 1 (because d q + 1) = (d − 1)qn−1 + dqn−2 + θq (n − 3). This completes the proof.
2
3.2. Example Our bound (1) is loose for n = 2, but is tight for n 3 even if we consider only irreducible hypersurfaces, as we see below. Lemma 3.1. Let X be a hypersurface of degree d in Pn over Fq . Regard this Pn as a hyperplane in Pn+1 defined by a linear equation over Fq , and take an Fq -point P 0 ∈ Pn+1 \ Pn . Then the cone X˜ over X with vertex P 0 is also a hypersurface of degree d in Pn+1 over Fq , and N q ( X˜ ) = qN q ( X ) + 1. Proof. If we choose coordinates X 0 , . . . , X n+1 of Pn+1 so that the hyperplane Pn is defined by X n+1 = 0 and P 0 = (0, . . . , 0, 1), the equation f ( X 0 , . . . , X n ) = 0 defining X in Pn also defines X˜ in Pn+1 . Since
X˜ = s(x0 , . . . , xn , 0) + t (0, . . . , 0, 1) f (x0 , . . . , xn ) = 0, (s, t ) ∈ P1 , we have N q ( X˜ ) = qN q ( X ) + 1.
2
Corollary 3.2. Under the assumption in Lemma 3.1, if N q ( X ) achieves the upper bound (1) in Pn , then so does N q ( X˜ ) in Pn+1 . Proof. Actually we have
N q ( X˜ ) = q (d − 1)qn−1 + dqn−2 + θq (n − 3) + 1
= (d − 1)qn + dqn−1 + qθq (n − 3) + 1 = (d − 1)qn + dqn−1 + θq (n − 2) by Lemma 3.1.
2
By this corollary, once an irreducible hypersurface over Fq in Pn achieves the upper bound (1), there exists one of the same degree in Pn+1 attaining the upper bound. So, to confirm the additional statement of Theorem 1.2, it is enough to consider surfaces in P3 .
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Example 3.3. Let Q be the image of the Segre embedding of P1 × P1 in to P3 over Fq , which is a hyperbolic quadric surface. In other words, under suitable choice of coordinates X 0 , . . . , X 3 of P3 , Q is defined by X 0 X 1 − X 2 X 3 = 0. Obviously, N q ( Q ) = (q + 1)2 = (2 − 1)q2 + 2q + θq (0). √
Example 3.4. Suppose q is a square number. It is known that the Hermitian surface H : X 0
q+1
+ √ 3 2 X1 + X2 + X3 = 0 in P has (q + 1)(q q + 1) Fq -points [1]. Since ( q + 1 − 1)q + √ √ ( q + 1)q + θq (0) = (q + 1)(q q + 1), N q ( H ) attains the upper bound (1). √
√
q+1
√
q+1
√
q+1
Example 3.5. Let F be a space filling surface q
q
q
q
X 0 X 1 − X 0 X 1 + X 2 X 3 − X 2 X 3 = 0. Then N q ( F ) = θq (3) = (q + 1 − 1)q2 + (q + 1)q + θq (0). Remark 3.6. Surfaces which appeared in the above Examples are nonsingular and in P3 , hence they are irreducible. 4. Comparison with other bounds 4.1. A bound from the Weil conjecture The Weil conjecture [13] established by Deligne [2] provides a bound for the number of Fq -points of nonsingular varieties defined over Fq , involving their Betti numbers. The Betti numbers b0 , b1 , . . . , b2(n−1) of a nonsingular hypersurface of degree d in Pn are the following [7, p. 163]:
• For i with 0 i < n − 1, bi = b2(n−1)−i = • Let b =
1 if i is even, 0 if i is odd;
bn−1 − 1 if n − 1 is even, 1 Then b = d− ((d − 1)n − (−1)n ). d bn−1 if n − 1 is odd.
Hence the Weil conjecture implies the following bound [2, Theorem 8.1]. Theorem 4.1 (Deligne). Let X be a nonsingular hypersurface of degree d in Pn defined over Fq . Then
N q ( X ) θq (n − 1) + b q
n −1 2
(4)
.
Now we compare our bound (1) with (4). Proposition 4.2. Suppose n 3. If d
√
q + 2, then
(d − 1)qn−1 + dqn−2 + θq (n − 3) < θq (n − 1) + b q
n −1 2
.
Proof. Let
F (x) =
=
x − 1 x
n −1 (x − 1)n − (−1)n q 2 + θq (n − 1) − (x − 1)qn−1 + xqn−2 + θq (n − 3)
x − 1 x
(x − 1)n − (−1)n − xq
n −1 2
− xq
n −3 2
q
n −1 2
+ qn−1 .
M. Homma, S.J. Kim / Finite Fields and Their Applications 20 (2013) 76–83
We want to prove that F (x) > 0 if x
√
q + 2. When x =
81
√
q + 2,
√
√ √ √ n −1 √ √ n −3 √ n −1 q + 1 √ n −1 √ ( q + 1)n − (−1)n − ( q + 2) q F ( q + 2) > √ q − ( q + 2) q + q q+2
√ √ n −2 √ n −3 √ n −1 q + 1 √ n −1 n n −1 −2 q + q + q =√ (n − 1) q q+2 2 3 n −4 n √ k n + q + 1 − (−1) k =1
k
√ √ n −3 q + 1 √ n −1 n n + −4 √ q n+ q > 0. q+2 2 3 Let F (x) be the derivative of F (x) by x. If x
F (x) =
(x − 1)n (nx + 1) − (−1)n x2
n −1
> ( x − 1)
−1−
√
n −1
q
−
√
−
√
q
√
q + 2, then
n −1
−
√
n −3
q
√
n −1
q
n−3 √ n−1
q
q
because
(x − 1)(nx + 1) x2
√ √ n −1 √ n −3 > 0. − q ( q + 1)n−1 − 1 − q Hence, for any x
√
q + 2, we have F (x) > 0.
>1
2
The following corollary was already remarked by Sørensen [10]. Corollary 4.3. If the number of Fq -points of a nonsingular hypersurface of degree d meets the bound (4) from √ the Weil conjecture, then d q + 1.
√
Proof. When n = 2, the bound (4) is N q ( X ) q + 1 + (d − 1)(d − 2) q, and the Sziklai bound says that N q ( X ) (d − 1)q + 1. Since
√
√
√
q + 1 + (d − 1)(d − 2) q − (d − 1)q + 1 = d − ( q + 1) (d − 2) q, any irreducible plane curve of degree d > statement is just the above proposition. 2
√
q + 1 cannot attain the bound (4). When n 3, this
4.2. Segre–Serre–Sørensen bound Without any restrictions on a hypersurface over Fq , the best bound was obtained by Serre [9], which is a generalization of Segre’s old result [8] for plane curves. Sørensen [10] also proved the same inequality as Serre’s.3 Theorem 4.4 (Segre–Serre–Sørensen). Let X ⊂ Pn be a hypersurface of degree d defined over Fq . Then N q ( X ) dqn−1 + θq (n − 2). Moreover, if d q, equality holds if and only if X is a union of d hyperplanes over Fq that contain a common linear subspace of codimension 2 in Pn .
3
The date on Serre’s letter to Tsfasman is 24 July 1989, and the received date of Sørensen’s paper is 17 January 1990.
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Remark 4.5. The assumption d q in the additional statement of the Segre–Serre–Sørensen theorem can be weakened as d q + 1 if n = 2 [8, II, §6, Observation IV], but cannot be when n 3 as Serre noted [9, Remark 2]. In the literature cited above, Serre also remarked that if a hypersurface X of degree d does not split into d Fq -hyperplanes, then a stronger bound than one in Theorem 4.4 holds:
N q ( X ) dqn−1 + θq (n − 2) − (q + 1 − d).
(5)
Our bound (1) is tighter than (5). In fact,
dqn−1 + θq (n − 2) − (q + 1 − d) − (d − 1)qn−1 + dqn−2 + θq (n − 3)
= (q + 1 − d) qn−2 − 1 .
4.3. Bounds of K. Thas In [12], Thas made refinements on Theorem 4.4. Those refinements involve a new invariant k X for a hypersurface X ⊂ Pn of degree d defined over Fq . The invariant k X is the maximal dimension of an Fq -linear subspace which is contained in X . He took two ways of finding an upper bound for N q ( X ); the first bound goes up according as k X increases, and the second goes down. Finally, he obtained a bound by mixing these two methods, whose assumption fits in ours. Theorem 4.6 (Thas). Let X be a hypersurface in Pn of degree d defined over Fq , and k = k X . If 0 < k n − 2, then
N q ( X ) dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1 . Moreover, if 0 < k < n − 2, strict inequality holds in (6). Remark 4.7. For the k = 0 case, he only showed a weak bound (5). Now we compare Thas’s bound (6) with ours (1). Proposition 4.8. For n 3, k n − 2 and d < q + 1, we have
(d − 1)qn−1 + dqn−2 + θq (n − 3) dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1 , and equality holds if and only if k = 0 or k = n − 2. It is easy to see that
dqn−1 + θq (n − 2) − (q + 1 − d) qk + qn−2−k − 1
= (d − 1)qn−1 + dqn−2 + θq (n − 3) + (q + 1 − d) qk − 1 qn−2−k − 1 .
This completes the proof.
2
(6)
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