Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes

We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in $\mathbb{P}^n(\mathbb{F}_q)$ of small degree $d$, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have applications to the weight distribution of the projective Reed-Muller codes PRM$(q,d,n)$ over the finite field $\mathbb{F}_q$.


Introduction
Let θ n = (q n+1 − 1)/(q − 1). Consider the set F q [X 0 , . . . , X n ] h d of all homogeneous polynomials of degree d over the finite field F q of order q in the n + 1 variables X 0 , . . . , X n . Consider also the n-dimensional projective space P n (F q ) over the finite field of order q, and order the points P 0 , . . . , P θn−1 of P n (F q ) in a certain way, where we normalize the coordinates of the points P i by making the leftmost non-zero coordinate equal to one.
The parameters of these linear codes are • Length of PRM(q, d, n) is θ n (see [3]).
For a more detailed introduction on the problem of determining the maximum number of points in varieties in projective spaces over finite fields, we refer the reader to [1,2,4].
For d ≤ q − 1, it is known that they correspond to the algebraic hypersurfaces which are the union of d hyperplanes passing through a common (n−2)-dimensional subspace of P n (F q ) [10].
A. Sboui also determined the second and third weight of PRM(q, d, n), with d < q and d > 7. He proved the following results [9, Theorem 3.10]. (2) The third weight of the code PRM(q, d, n) is defined by the algebraic hypersurfaces A d 3 of degree d which are the union of d hyperplanes, d − 2 of which meet in a common subspace K 1 of dimension n − 2, and where the last two hyperplanes H d−1 and H d meet in a subspace K 2 , different from K 1 , such that K 2 is contained in exactly one of the d − 2 hyperplanes passing through K 1 .
A. Sboui also determined the configuration of d (< q) hyperplanes H 1 , . . . , H d having the smallest number of points. This configuration is described in the following way: for every subspaces of dimension n − 2, all distinct and meeting in a common subspace of dimension n − 3. He furthermore proved that if q > d(d − 1)/2, then any algebraic hypersurface of degree d, not the union of d hyperplanes, contains fewer points than any algebraic hypersurface which is the union of d hyperplanes.
The consequence of this result is the following: the weight w l m given by the minimal hyperplane arrangement is the highest weight of a codeword in PRM(q, d, n) which can be given by any hyperplane arrangement. Moreover, for q > d(d − 1)/2, any algebraic hypersurface of degree d in P n (F q ) containing an absolutely irreducible non-linear factor cannot correspond to a codeword with weight less than w l m , see [6]. If we suppress the condition q > d(d − 1)/2, Rodier and Sboui proved that the third highest number of points, so the third weight w 3 , given by an A d 3 hyperplane arrangement, is also reached by a hypersurface of degree d composed of d − 2 hyperplanes and one irreducible quadric hypersurface. We continue the study of Sboui for d < 3 √ q. The goal of this paper is to determine results related to the question how many points an algebraic curve (resp. an algebraic hypersurface) over a finite field can have, depending on its number of lines (resp. hyperplanes).
In this way, also this article contributes to the determination of the weights of the d-th order q-ary projective Reed-Muller codes PRM(q, d, n), for small order d [3,11].

Algebraic plane curves
In this article, we will make regularly use of the Hasse-Weil bound for an absolutely irreducible algebraic curve defined over F q . Theorem 2.1 (Hasse and Weil). Let X be an absolutely irreducible algebraic curve of degree d defined over F q , then The following lemma presents an upper bound on the number of points of an algebraic plane curve, not containing a linear component defined over F q . The lower bound q > 13 in the next lemma arises from the calculations for the upper bounds on the size of the algebraic hypersurface Φ in Subsection 4.1. There we use that for q > 13, 2 2 (q + 1) + 2 √ q, for d odd, d 2 (q + 1), for d even. Proof. We proceed by induction on d.
The result is true for d = 2 and for d = 3. Assume that the formula is valid for all d ≤ d − 1.
(1) If C contains a conic C, then C = C ∪C, #C = q +1, and we use the inequality #C ≤ #C + #C.
(1.1) If the degree d of C is even, we apply the induction hypothesis to C , and then #C ≤ d 2 (q + 1).
(2) If C contains an absolutely irreducible cubic curve C; using the Hasse-Weil bound for an absolutely irreducible cubic curve (#C ≤ q + 1 + 2 √ q), we prove the result by the same method as the case (1). (3) We now study the case where C does not contain a conic nor an absolutely irreducible cubic curve. In this case, C only contains absolutely irreducible components of degree m, 4 ≤ m ≤ d. Let be the number of absolutely irreducible components C i of odd degree m i in C, and let k be the number of absolutely irreducible components C i of even degree n i in C, Now the following inequalities are valid, see also [13]: Since 2 √ q − q+1 2 < 0, we conclude with the two following cases: a) If d is even, is also even. From inequality (1) Remark 1. The bounds of Lemma 2.2 are sharp for d even. For d ≤ 2(q − 1) even, it is possible to construct d/2 pairwise disjoint conics. For q odd, the conics C c : Remark 4.2]. Similarly, for q even, in [15,Remark 3], it is proven that the conics C f : X 2 0 + f d 1 X 2 1 + f X 2 2 + X 1 X 2 = 0, are pairwise disjoint when f ∈ F * q and Tr(d 1 ) = 1 with Tr the trace function from F q to F 2 .
For d even, using d/2 of these pairwise disjoint conics, an algebraic curve of degree d containing d(q + 1)/2 points can be constructed. Lemma 2.3. Let C be an algebraic plane curve of degree d in P 2 (F q ), such that 2 ≤ d ≤ √ q 2 and q > 13. If C contains at most r different lines defined over F q , then #C ≤ B r , where Proof. We proceed by induction on d. For d = 2, the only possible values for r are 0, 1, or 2, and the result is true. Assume that the result is true for all the degrees less than or equal to d − 1. Let C be an algebraic curve of degree d containing at most r lines (0 ≤ r ≤ d).
1) For r = 0, C does not contain any line, so from Lemma 2.2: #C ≤ B, where which satisfies #C ≤ B 0 .
2) Assume that C contains only one line (r = 1), then C = C ∪ ∆, ∆ a line and C an algebraic curve of degree d − 1. Then #C ≤ #C + #∆; we apply Lemma 2.2 to C , and we obtain that #C ≤ B, where which is equal to B 1 .
Theorem 2.4. Let C be an algebraic plane curve of degree d over F q , such that 2 ≤ d ≤ √ q 2 and q > 13. If #C > B r−1 , then C contains at least r different lines defined over F q .
Proof. We can deduce the result directly from Lemma 2.3 by rewriting the bound B r for r − 1. Indeed, is an upper bound on the size of such algebraic plane curves containing at most r −1 different lines defined over F q . The result follows from the fact that the lower bound B r−1 , for constant d, is an increasing function in r, i.e., B r > B r−1 , independent of the parity of d − r.

The existence of a d-secant not containing any singular point of the algebraic hypersurface Φ
The goal is now to use the preceding upper bounds on the number of points of algebraic curves of degree d to determine upper bounds on the number of points of algebraic hypersurfaces Φ of degree d < 3 √ q in P n (F q ) containing exactly r − 1 different linear components H 1 , . . . , H r−1 defined over F q . To derive these upper bounds, we need to have a particular d-secant to this algebraic hypersurface Φ. The following theorems prove the existence of such a particular d-secant to Φ.
Theorem 3.1. Consider an absolutely irreducible algebraic hypersurface Φ : F (X 0 , . . . , X n ) = 0 in P n (F q ), where deg F = d, d < 3 √ q, then through a point of P n (F q ) not on Φ, there is a line of P n (F q ) not containing a singular point of Φ.
Assume that ∂F ∂X0 (X 0 , . . . , X n ) ≡ 0. To find an upper bound on the number of singular points of Φ, we calculate the resultant R(F, ∂F ∂X0 ) of F (X 0 , . . . , X n ) and ∂F ∂X0 (X 0 , . . . , X n ) with respect to X n . Assume that R(F, ∂F ∂X0 ) has degree e ≤ d(d − 1). Note that the resultant R(F, ∂F ∂X0 ) cannot be identically zero, or else ∂F ∂X0 divides F , which is impossible.
We also need to avoid the, at most, intersection points of the r − 1 hyperplanes contained in Φ. Hence, we need to avoid at most d 3 2 q n−2 + d 2 2 θ n−2 < q n−1 + · · · + q + 1 points, so indeed, there is a line through every point not on Φ containing none of the singular points of any of the components of Φ, and intersecting the r − 1 hyperplanes of Φ in r − 1 distinct points.
4. Hypersurfaces in P n (F q ) 4.1. Determination of the upper bounds. We now wish to obtain an upper bound on the number of points in P n (F q ) belonging to an algebraic hypersurface Φ of degree d < 3 √ q, containing r − 1 distinct hyperplanes defined over F q .
By Theorem 3.2, we know that there exists a line in P n (F q ), not contained in Φ, not containing any of the singular points of Φ, and intersecting the r − 1 hyperplanes of Φ in r − 1 distinct points. Assume that P 1 , . . . , P d are the d points of belonging to Φ, counted according to the intersection multiplicities of and Φ in their intersection points. We can partition these points P 1 , . . . , P d into the following sets: (1) the set of points {P 1 , . . . , P r−1 } belonging to the linear components of Φ, (2) the set of points {P r , . . . , P m } all belonging to P n (F q ) and belonging to the non-linear components of Φ, and (3) the set of points {P m+1 , . . . , P d } not belonging to P n (F q ), but to an extension of P n (F q ), and again belonging to the non-linear components of Φ.
Assume that exactly m ≤ m of the points P 1 , . . . , P m are distinct. We form this partition for the following reasons. First of all, the points P 1 , . . . , P r−1 belong to a linear component, i.e. a hyperplane, of Φ, hence, they belong to θ n−2 lines of Φ in these hyperplanes. Secondly, the points P i , i ∈ {r, . . . , m}, belong to a non-linear component of Φ of degree d i > 1. Here, P i is a non-singular point of this non-linear component, so has a tangent hyperplane Π i to this component. All the lines of Φ through P i lie in this tangent hyperplane Π i . This tangent hyperplane is not contained in Φ, so contains at most d i q n−2 + θ n−3 points of Φ [10], so this implies that P i belongs to at most d i q n−3 + θ n−4 lines of Φ.
Consider this line and consider all θ n−2 planes of P n (F q ) through . Denote them by H 1 , . . . , H θn−2 . Assume that H i ∩ Φ contains x i lines defined over F q . It is possible that some of these x i lines coincide. This will be no problem for the future arguments. To have a correct upper bound on the number of points of the algebraic curves H i ∩Φ, it is no problem to assume that these x i lines are all pairwise distinct. Let H 1 , . . . , H k be the planes through for which d − x i is odd. Then we obtain the following upper bound on the size of Φ: The last term is negative for q > 13. We now discuss separately the two cases d − r + 1 odd and d − r + 1 even.