Incidences of Cubic Curves in Finite Fields

In this paper we prove an incidence bound for points and cubic curves over prime fields. The methods generalise those used by Mohammadi, Pham, and Warren (2021).


Introduction
Given a set of points P in the plane F 2 over a field F, and a set of irreducible algebraic curves C in F 2 , the number of incidences between P and C is defined as In the case F = R and when C is actually a set of lines L, an optimal upper bound for I(P, L) was given by Szemerédi and Trotter [11].
Over R, this theorem has been generalised to other curves, the most well known such result being the Pach-Sharir theorem, see [5], [6].Such results for algebraic curves have also been proven over the complex numbers, see [9].
In this paper we consider the case F = F p for prime p.In this setting, point-line incidence bounds analogous to Theorem 1 are known, the first such result being proved by Bourgain, Katz, and Tao [1].The state of the art point-line incidence bound is due to Stevens and de Zeeuw [10] which itself relies on the point-plane incidence bound of Rudnev [8].Given that the sets of points and lines are not too large with respect to the characteristic p, they give the bound Using the basic geometric fact that two lines intersect in one point, and two points define one line, one can apply the Kővári-Sós-Turán theorem [3] to the incidence graph of P and L to obtain The bound (1) improves upon these bounds for a certain balancing of |P | and |L|.
Obtaining incidence bounds between points and non-linear algebraic curves in F p has proved a difficult task, with very few results being known.However, recently there has been a flurry of activity concerning incidences between points and certain degree two curves in F p , see for instance [7] and [12].Pushing the methods used in these papers further, an incidence bound between points and arbitrary irreducible conics was given in a paper of Mohammadi, Pham, and Warren [4].
In this paper, we adapt and generalise ideas present in [4] to prove an incidence bound between points and arbitrary cubic curves in F p .Our main result is the following.It is again important to compare this result with the trivial bounds given by Kővári-Sós-Turán.As above, this is given by the basic fact that two irreducible cubic curves intersect in at most nine points.This yields Comparing these bounds to the first term in Theorem 3, we see that Theorem 3 improves upon the trivial bounds when we have and within this range the second term of Theorem 3 is dominated by the first.Theorem 2 is then the augmentation of Theorem 3 with the Kővári-Sós-Turán bounds.We note that although we have focused on F p , the results extend to other fields, with the same restriction on the size of P with respect to the characteristic p, and also to fields of characteristic zero by ignoring the restriction on the characteristic.
We mention that it is crucial to restrict to irreducible curves in Theorem 2 (and such incidence results in general), as otherwise I(P, C) = |P ||C| is obtainable.Take a single line l, and let all of P lie on l.Define a set of reducible cubic curves C, where each is the union of l with some other conic.Since every point lies on l, which is a component of every cubic in C, the number of incidences is |P ||C|.

Proof of Theorem 3 2.1 The set-up
We now begin the proof of Theorem 3. The main idea will be to, in a certain sense, dualise the points and curves P and C, so that we recover point and line incidences.However, we will not work with incidences directly, choosing to instead work with k-rich curves.A curve γ ∈ C is called k-rich if it contains between k and 2k points of P , that is, We let C k ⊆ C be the set of k-rich curves from C. Our main goal will be to bound, for all k sufficiently large, |C k |.This will be achieved by first considering the problem locally.
Let S ⊆ P be a set of seven points.We make the definition In words, this is the set of k-rich curves which pass through all points of S. Given a bound for each C k,S , we can give a bound on C k .Indeed, if we sum over all subsets S ⊆ P of size seven, each k-rich curve will be counted at least k 7 ≫ k 7 times, noting that this assumes k ≥ 7.This implies that we have the inequality We now begin the main part of the proof, which is to bound |C k,S |.

Bounding C k,S
To begin the dualisation process, we provide a map φ which sends our curves C to points in P(F 9 p ).The map is very simple -it takes a curve f (x, y) = 0 to its list of coefficients.Note that this is a map into projective space since constant multiples of an equation f (x, y) = 0 determine the same curve.
φ : {Curves of degree at most 3 over The ordering chosen for the coordinates is irrelevant -we simply fix an ordering and use it consistently.
Fix a point q = (q 1 , q 2 ) ∈ F 2 p .If we let Γ q be the set of all degree at most 3 curves passing through q, then the image φ(Γ q ) is a hyperplane in P(F 9 p ), since the point q imposes a single linear condition on the coefficients of the curves.Indeed, the points [X 0,0 : X 0,1 : ... : X 2,1 : X 3,0 ] ∈ φ(Γ q ) are precisely those that satisfy the linear equation We denote such a hyperplane by π q .We now take our set S ⊆ P of size seven, and look at the image under φ of all degree at most 3 curves which pass through the points of S, call them Γ S .From the above, this is given by φ(Γ S ) = q∈S π q .
We prove a lemma to control this image.
Lemma 1.Let S ⊆ P be a set of seven points.Then either φ(Γ S ) is a 2-flat, or C k,S is the empty set.
In order to prove this, we require a simple proposition.The following is a version of a result present in [2] -a proof can be found there which is valid over sufficiently large fields.Proposition 1.Let S be a set of points in F 2 p .
• If |S| = 7 and S contains no five collinear points, then S imposes independent conditions on the set of all cubic curves.
• If |S| = 8 and S contains no five collinear points and are not all one a common conic, then S imposes independent conditions on the set of all cubic curves.
The statement "S imposes independent conditions on the set of all cubic curves" means that the intersection ∩ q∈S π q is complete, that is, has dimension two.Note that the only way this can fail to happen is if at some point one of these intersections were trivial, that is, a hyperplane π q contains the previous intersections ∩ q ′ ∈S ′ π q ′ for some subset S ′ ⊂ S. If this happens, then every cubic curve passing through all of S ′ also passes through q.We can now prove Lemma 1.
Proof of Lemma 1.Note that if S were contained in a conic, we must have C k,S = ∅, as otherwise this conic intersects an irreducible cubic curve in seven points.This implies that Γ S contains only cubic curves.If S contains four collinear points, then S cannot be contained within any irreducible cubic curve, by Bezout's theorem, and therefore C k,S = ∅.On the other hand, if no four points of S are collinear, then by Proposition 1, the intersections of the hyperplanes π q for q ∈ S is complete, so that φ(Γ S ) is a 2-flat.
We continue the proof, assuming that |C k,S | = 0, implying that φ(Γ S ) is a 2-flat.Let π S denote this 2-flat.We have that φ(C k,S ) ⊆ π S , and π S contains only points corresponding to cubic curves.The next step is to give a map which sends our original points P to lines in π S .Since points not lying on any curve from C k,S do not contribute any incidences, we only perform this step for points which do indeed lie on curves from C k,S -by an abuse of notation we denote such points by P ∩ C k,S .Furthermore, we ignore the points of S, as they would be, in a certain sense, degenerate for this map.We define the map as follows.
We must justify, firstly, that ψ(q) is indeed a line in π S .Since we are intersecting a hyperplane with a 2-flat, ψ(q) can either be a line, as needed, or we have π q ∩ π S = π S .If this second case were to occur, it would mean that π S ⊆ π q , so that S ∪ {q} does not impose independent conditions on cubic curves, which by Proposition 1 implies that it contains five collinear points, or all eight are on a conic.In the first case, by removing q we find at least four points of S collinear, contradicting the assumption |C k,S | = 0.In the second case, we must have that S lies on a conic, again contradicting Bezout's Theorem unless C k,s = ∅.We therefore conclude that ψ(q) is indeed a line.
Secondly, we check the multiplicity of the lines ψ(q).We claim that for each line l lying in π S , there are at most two points q, q ′ which are both mapped to l, that is, these lines are defined with multiplicity at most two.To prove this, suppose there exist three points q 1 , q 2 , q 3 with ψ(q 1 ) = ψ(q 2 ) = ψ(q 3 ) =: l.Consider the set S ∪ {q 1 , q 2 , q 3 }.Since q 1 , q 2 , q 3 ∈ (P ∩ C k,S ) \ S, there must exist an irreducible cubic curve γ ∈ C k,S such that φ(γ) ∈ l.Indeed, this follows since we have for all q ∈ (P ∩ C k,S ) \ S, and γ ∈ C k,S , Then γ contains the ten points S ∪ {q 1 , q 2 , q 3 }.On the other hand, since l is a line, we can take any point other than φ(γ) on l, and we find another (possibly reducible) cubic curve containing S ∪ {q 1 , q 2 , q 3 }.Since γ is irreducible, this contradicts Bezout's theorem.
We now put together all of the above information, to recover an incidence problem between points and lines in F 2 p .Take a k-rich curve γ ∈ C k,S .It has been mapped to a point φ(γ) ∈ π S .Each point q ∈ P \ S which lies on γ has been sent, via ψ, to a line ψ(q) ⊆ π S , and this line must contain the point φ(γ), since q ∈ γ.Such lines are defined with multiplicity at most two.Therefore, the k-rich curve γ has been sent to an at least k−7 2 -rich point φ(γ), with respect to the lines L := ψ((P ∩ C k,S ) \ S).

Finishing the proof
Returning to equation (2), we can bound the number of k-rich curves for k ≥ 11 as We can now follow a standard argument to bound I(P, C).In the following we denote by C =k the set of precisely k-rich curves.
We can now bound |C k,S | by the number of k−7 2 -rich points defined by a set of |L| ≤ |P | lines in F 2 p ∼ = π S .This is done via the following result of Stevens and de Zeeuw [10].Corollary 1.Let L be a set of lines in F 2 p , with |L| ≪ p 15/13 , and for t ≥ 2 let P t denote the number of t-rich points with respect to L. Then (P, C) = 9+3/4 (2 i ∆) 10+3/4 + |P | 8 (2 i ∆) 8 (2 i ∆)We now optimise our choice of ∆.In order to ensure that the application of Corollary 1 was valid, we must have ∆ ≥ 11.The best choice is then∆ = max 11, |P | 39/43 |C| 4/43 .If the second term is taken in this maximum, we recover the first two terms of Theorem 3. If the first term is chosen, then we must have |C| 4 ≫ |P | 39 , and in this case our bound gives I(P, C) ≪ |C|.Combining these two possibilities yields Theorem 3.