Finite Fields and Their Applications

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1 Finite Fields and Their Applications ) Contents lists available at ScienceDirect Finite Fields and Their Applications Sziklai s conjecture on the number of points of a plane curveoverafinitefieldiii Masaaki Homma a,,1, Seon Jeong Kim b,2 a Department of Mathematics, Kanagawa University, Yokohama , Japan b Department of Mathematics and RINS, Gyeongsang National University, Jinju , Republic of Korea article info abstract Article history: Received 9 February 2010 Revised 19 May 2010 Available online 1 June 2010 Communicated by Neal Koblitz MSC: 14H50 14G15 14G05 14N10 We manage an upper bound for the number of rational points of a Frobenius nonclassical plane curve over a finite field. Together with previous results, the modified Sziklai conjecture is settled affirmatively Elsevier Inc. All rights reserved. Keywords: Plane curve Finite field Rational point Frobenius nonclassical curve 1. The modified Sziklai conjecture The goal of this paper is to settle the modified Sziklai conjecture, which has been the motivation for a series of our works [4 6]. Let C be a plane curve of degree d over a finite field F q without F q -linear components. Even though an F q -point of P 2 is a singular point of C, it is counted in an F q -rational point of C, in other words, we are not talking about the branches of a singular point. The * Corresponding author. addresses: homma@n.kanagawa-u.ac.jp M. Homma), skim@gnu.kr S.J. Kim). 1 Partially supported by Grant-in-Aid for Scientific Research ), 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 ) /$ see front matter 2010 Elsevier Inc. All rights reserved. doi: /j.ffa

2 316 M. Homma, S.J. Kim / Finite Fields and Their Applications ) set of F q -rational points of C is denoted by CF q ), and the cardinality of CF q ) by N q C). Weare interested in finding a good bound for N q C) by means of d and q. When C is absolutely irreducible, people frequently consider the number N q C) of Fq -rational points of the normalization C of C, however the inequality Nq C) Nq C) is not always true. Namely we can t use a bound for N q C) directly in our context. Modified Sziklai Conjecture. Unless C is a curve defined over F 4 which is projectively equivalent to over F 4, the inequality X 4 + Y 4 + Z 4 + X 2 Y 2 + Y 2 Z 2 + Z 2 X 2 + X 2 YZ+ XY 2 Z + XY Z 2 = 0 1) would hold true. N q C) d 1)q 2) This conjecture was originally posed by Sziklai [8] in the form that 2) might have held without exception, however he had missed the counter-example 1). Since the cardinality of the set of F q - points of the ambient plane P 2 is q 2 + q, the conjecture makes sense in the range 2 d q. Since it is easy to see that N q C) d 1)q if C is reducible [5, Propositions 2.1 and 2.2], we may suppose that C is irreducible in order to verify the conjecture, and do so hereafter. In the previous works, we observed the conjecture to be true if i) d = q [4, Corollary 2.2] or d = q [5, Theorem 1]; or ii) C is q-frobenius classical [5, Proof of Theorem 4.1 and Remark 4.2]; or iii) C is nonsingular and q-frobenius nonclassical [5, Theorem 4.1 and Remark 4.2]. Therefore, to settle the conjecture, it is enough to consider only q-frobenius nonclassical plane curves of degree d with 2 d q 1. The properties of a q-frobenius nonclassical plane curve necessary for the proof are explained in the next section. For full details, see [3, Chapters 8 and 9]. 2. The q-frobenius nonclassical plane curve Let p be the characteristic of F q, and q = p e. Additionally, let F X, Y, Z) be a reduced homogeneous polynomial over F q whose zero set is an irreducible plane curve C of degree d. The plane curve C is q-frobenius nonclassical if the polynomial F X, Y, Z) divides F X X q + F Y Y q + F Z Z q, where F X, F Y and F Z are partial derivatives by X, Y and Z respectively. Geometrically this notion means that for a general point Q C the q-frobenius image Q q) of Q, which is the point taking coordinatewise q-th power of Q, lies on the tangent line T Q C) to C at Q. In this case, the intersection multiplicity ic.t Q C); Q ) of C and T Q C) at Q is a power of p, sayp i,withe i 1 see [1, Propositions 1, 3 and 4] or [2, 2]). In terms of divisors on C, for a general point Q C, If P C is an F q -rational nonsingular point, then C.T Q C) p i Q + Q q). 3) i C.T P C); P ) p i, 4) which can be understood intuitively by taking a limit Q P in 3). A rigorous proof of the fact 4) can be found in [7, Corollary 2.6]. The following lemma is a specialized form of [4, Theorem 2.1] in the Frobenius nonclassical curve.

3 M. Homma, S.J. Kim / Finite Fields and Their Applications ) Lemma 2.1. Let C be a q-frobenius nonclassical curve of degree d over F q. Suppose that all the F q -rational points are nonsingular. Then N q C) q p i + p2i + p i ) d. q + p i Proof. Consider the point-line correspondence P = { P,l) CF q ) ˇP 2 F q ) P l }, with two projections π 1 : P CF q ) and π 2 : P ˇP 2 F q ), where ˇP 2 F q ) is the projective plane of F q -lines in the original projective plane. For a fixed l ˇP 2 F q ), let l CF q ) ={P l, j } j Jl. Since j J l il.c; P l, j ) l.c) = d, # π 1 2 l) = # l CF q ) ) = # J l d j J l il.c; Pl, j ) 1 ), where l.c) is the intersection number of l and C. Hence # P = l ˇP 2 F q ) # π 1 2 l) # ˇP 2 F q ) d l ˇP 2 F q ) j J l = q 2 + q ) d il.c; Pl, j ) ) 1 P CF q ) l ˇP ) il.c; P) 1, where ˇP denotes the set of Fq -lines passing through P CF q ).ForeachpointP CF q ), ic.l; P) p i ifl = T P C) by 4), and ic.l; P) = 1forl ˇP \{TP C)} because P is nonsingular. Therefore # P q 2 + q )d N q C) p i. On the other hand, counting the cardinality of P by using π 1,we have # P = N q C) q ). Hence N q C) q + p i ) q 2 + q ) d = q p i) q + p i ) + p 2i + p i ) d. This completes the proof. Another bound for N q C) of a q-frobenius nonclassical curve C is known. Lemma 2.2. Under the same assumption as in Lemma 2.1, N q C) 1 2 d p i d 3) + q + 2 ). Proof. Let C C be the normalization of C, and g the genus of C. Then 1 N q C) p i 2g 2) + q + ) 2)d. 2 This inequality is a special case of [7, Theorem 2.13], or a direct proof of it for the plane curve case can be found in [2, Theorem 1.3]. Since g p a C) = 1 2 d 1)d 2) and CFq ) = CF q ), the desired inequality is proven.

4 318 M. Homma, S.J. Kim / Finite Fields and Their Applications ) Theorem 2.3. Let q be a power of a prime number p, and say q = p e. Let C be a q-frobenius nonclassical irreducible curve of degree d over F q,andp i the intersection multiplicity ic.t Q C); Q ) for a general point Q C. Then N q C) d 1)q ; and moreover N q C) d 1)q ifd p e i. Proof. As was explained in the previous work [5, Proposition 2.3], N q C) d 1)q if C has a singular point which is an F q -point. So we may assume that all the F q -points of C are nonsingular. We divide the proof into three cases. a) Suppose that 2 d p e i. Compare the bound in Lemma 2.2 with the desired one; d 1)q 1 2 d p i d 3) + q + 2 ) = 1 2 dq q d 1 2 pi dd 3) = 1 2 d 2)q 2) pi dd 2) 2 pi d = 1 2 d 2) q 2 p i d ) 2 pi d 2. 5) If p i d q 2, then 5) is nonnegative because d 2 and i > 0. Hence we have the desired inequality when d p e i 2 p, in particular for d i pe i 1. When d = p e i,5)isequalto 1 2 pe p e i, which is nonnegative. b) Suppose that p e i + 2 d. Compare the bound in Lemma 2.1 with d 1)q ; d 1)q q p i + p2i + p i ) q + p i d 1 = q 2 q + p i qp i + p i + qp i ) ) 1 d 1 qp i + q + p i p i p e i ) 1 because d p e i + ) 2 ) ) 1 = q + p i q p i 1pi + p i 1 > 0. Hence N q C) d 1)q. c) Lastly we consider the case d = p e i. In this case, the upper bound for N q C) in Lemma 2.1 is equal to p e p i + p2i + p i ) p e i + ) p e + p i 1 = p 2e i + pe + p e i p e + p i Since p e i p i < p e i < p e + p i, we know means that = p 2e i + pe i p i p e + p i. pe i p i p e +p i +1 < 1. Hence, when d = pe i, Lemma 2.1 N q C) p 2e i = d 1)q. This completes the proof. As was mentioned in the previous section, from Theorem 2.3 we can conclude the modified Sziklai conjecture to be true.

5 M. Homma, S.J. Kim / Finite Fields and Their Applications ) Conclusion In this paper, we established the following theorem. Theorem 3.1. If C is a plane curve of degree d 2 over F q without F q -linear components, then the number of F q -point N q C) is bounded by except for the curve over F 4 defined by Eq. 1). N q C) d 1)q, References [1] A. Hefez, J.F. Voloch, Frobenius nonclassical curves, Arch. Math. Basel) ) ; correction: Arch. Math. Basel) ) 416. [2] J.W.P. Hirschfeld, G. Korchmáros, On the number of solutions of an equation over a finite field, Bull. Lond. Math. Soc ) [3] J.W.P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic Curves over a Finite Field, Princeton Univ. Press, Princeton, NJ, [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 ) [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, AMS, Providence, RI, 2010, in press; an earlier version is available at arxiv: [6] M. Homma, S.J. Kim, Toward determination of optimal plane curves with a fixed degree over a finite field, preprint, [7] K.-O. Stöhr, J.F. Voloch, Weierstrass points and curves over finite fields, Proc. Lond. Math. Soc. 3) ) [8] P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl )