Intersection of two quadrics with no common hyperplane in $\mathbb{P}^{n}(\mathbb{F}_q)$}}

TL;DR

This paper establishes the tightest known upper bound on the number of rational points in the intersection of two quadrics in projective space over finite fields, and proposes a conjecture for algebraic sets of higher degree.

Contribution

It provides the best upper bound for the intersection points of two quadrics with no common hyperplane and proposes a general conjecture for algebraic sets.

Findings

Upper bound: |Q1 ∩ Q2| ≤ 4q^{n-2} + π_{n-3}

Conjecture: |X(𝔽_q)| ≤ d q^s + π_{s-1} for algebraic sets

Inspiration for future bounds on algebraic set points

Abstract

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that $| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+\pi_{n-3}$. This result inspires us to establish the conjecture on the number of points of an algebraic set $X\subset {\mathbb{P}}^n(\mathbb{F}_q)$ of dimension $s$ and degree $d$: $|X(\mathbb{F}_q)|\le dq^s+\pi_{s-1}$.

Equations

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 2(4q^{n-2}+4{\pi }_{n-3})+\frac{7}{q-1}.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 2(4q^{n-2}+{\pi }_{n-3})+\frac{1}{q-1}.$$

$$LS(n,q)=\{\begin{array}{ll}2q^{n-2}+{\pi }_{n-3}+2q^{\frac{n-1}{2}}-q^{\frac{n-3}{2}}& \text{if}n+1\ge 4\text{and}\text{even}\\ 2q^{n-2}+{\pi }_{n-3}+q^{\frac{n}{2}}& \text{if}n+1\ge 5\text{and}\text{odd}.\end{array}$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 4q^{n-2}+{\pi }_{n-3}$$

$$|X({\mathbb{F}}_q)|\le d{q}^s+{\pi }_{s-1}.$$

$${\pi }_n=\#{\mathbb{P}}^n({\mathbb{F}}_q)=q^n+q^{n-1}+...+1.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le m{q}^l+{\pi }_{l-1}.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 4q^{n-2}+{\pi }_{n-3}$$

$$\mathrm{for}4\le i\le n-1,|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 4q^{i-2}+{\pi }_{i-3}.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2\cap {\mathbb{E}}_{n-l}({\mathbb{F}}_q)|\le 4q^{n-l-2}+{\pi }_{n-l-3}.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 4q^{n-2}+{\pi }_{n-3}.$$

$$|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le EH(n,q)$$

$$EH(n,q)=\{\begin{array}{ll}2q^{n-2}+{\pi }_{n-3}+2q^{\frac{n-1}{2}}-q^{\frac{n-3}{2}}& \text{if}n+1\ge 4\text{and}\text{even}\\ 2q^{n-2}+{\pi }_{n-3}+q^{\frac{n-3}{2}}& \text{if}n+1\ge 4\text{and}\text{even}\\ 2q^{n-2}+{\pi }_{n-3}+2q^{\frac{n-2}{2}}& \text{if}n+1\ge 5\text{and}\text{odd}.\end{array}$$

$$|{\Pi }_{n-r}{\mathcal{Q}}_{r-1}\cap \mathcal{Q}|\le EH(r-1,q)q^{n-r+1}+{\pi }_{n-r}.$$

$$|X({\mathbb{F}}_q)|\le d{q}^s+{\pi }_{s-1}.$$

$$|X({\mathbb{F}}_q)|\le d{\pi }_s.$$

$$|X({\mathbb{F}}_q)|\le d({\pi }_s-{\pi }_{2s-n})+{\pi }_{2s-n}.$$

Full text

Intersection of two quadrics with no common hyperplane in $\mathbb{P}^{n}(\mathbb{F}_{q})$

$\text{Fr\'{e}d\'{e}ric A. B. Edoukou}^{*}$, $\text{San Ling}^{*}$, Chaoping Xing

Division of Mathematical Sciences,

Nanyang Technological University,

21 Nanyang Link, Singapore 637371.

E.mail : $\{\mathrm{abfedoukou,lingsan,xingcp}\}$@ntu.edu.sg This work is supported by MOE-AcRF Tier 2 Research Grant, Singapore (No.T206B2204).

Abstract

Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^{n}(\mathbb{F}_{q})$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}|\leq 4q^{n-2}+\pi_{n-3}$. This result inspires us to establish the conjecture on the number of points of an algebraic set $X\subset{\mathbb{P}}^{n}(\mathbb{F}_{q})$ of dimension $s$ and degree $d$: $|X(\mathbb{F}_{q})|\leq dq^{s}+\pi_{s-1}$.

Keywords: finite fields, quadric varieties.

Mathematics Subject Classification: 05B25, 11T71, 14J29

1 Introduction

In 1954, L. Carlitz [4, Theorem 5, p.144] gave a quite formula to compute the number of points in the intersection of two quadrics. His formula depended mainly on the capacity to write the first quadric as a sum of non-degenerate quadrics $\mathcal{Q}_{i}$, the number of indeterminates of each quadric $\mathcal{Q}_{i}$, and the number of quadrics $\mathcal{Q}_{i}$ with odd indeterminates. Secondly on the capacity to write the second quadrics as linear combination of the quadrics $\mathcal{Q}_{i}$.

In the same year A. Weil [17, p.348] extend the result of L. Carlitz, to two arbitrary quadrics. His formula depended on the ability to diagonalize the forms in the pencil defined by the two quadrics. We think that these sophisticated methods are concretly difficult to apply, in order to find an interesting upper bound for the number of points in the intersection of two quadrics.

In 1975, W. M. Schmidt [14., Lemma 3C, p. 175], was the first the give an explicit upper bound for the number of intersection of two hypersurfaces by using some results on the theory of resultants. Therefore for two quadrics, his estimate leads to the following result:

[TABLE]

But he was convinced that its estimate was not the best possible.

In 1986, A. A. Bruen and J. W. P. Hirschfeld [3, pp. 218-220] computed the number of points in the intersection of two quadrics only for non-degenerate quadrics in a projective space of odd dimension. Even in these cases, theirs results don’t satisfy totally our hope.

In 1992, Y. Aubry [1, Theorem 3, p. 11] inspired by the techniques of W. M. Schmidt and the upper bound for hypersurfaces of Tsfasman-Serre-Sørensen [15], [16, chap.2,pp.7-10], improved the above upper bound. We prefer to write his result in this way:

[TABLE]

In 1999, D. B. Leep and L. M. Schueller [13, p.172] improved the result of Y. Aubry under the condition that the pair of quadrics have full order. This is a too much restrictive condition. These are their bounds: $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}|\leq LS(n,q)$ with

[TABLE]

In 2008, F. A. B. Edoukou [6] gave the best upper bound for the number of points in the intersection of two quadrics with no common plane in ${\mathbb{P}}^{3}(\mathbb{F}_{q})$, when one of them is non-degenerate (hyperbolic or elliptic) or degenerate of rank 3. He deduced that $4q+1$ is the optimal bound.

In 2009, F. A. B. Edoukou [7] study the same problem in ${\mathbb{P}}^{4}(\mathbb{F}_{q})$ when one of the two quadrics is non-degenerate quadric (i.e. parabolic), degenerate of rank 4 or 3. He deduced that $4q^{2}+q+1$ is the best upper bound and he formulated the following conjecture:

Conjecture [7, p.31]: Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ two quadrics in ${\mathbb{P}}^{n}(\mathbb{F}_{q})$ with no common hyperplane. Then

[TABLE]

And this bound is the best possible.

In this paper we will give a proof of this conjecture. The paper has been organized as follows. First of all we recall some generaties on quadrics. Secondly, based on what has been done for the study of intersection of quadrics in the projective of three and four dimension, and section of two quadrics with full order in higher dimension, we prove the conjecture. Finally we will formulate another conjecture for the intersection of quadrics, and a generalization of the above conjecture for a projective algebraic set of dimension $s$ and degree $d$ in a conjectural way:

[TABLE]

We will discuss on this conjecture.

2 Generalities

We denote by $\mathbb{F}_{q}$ the field with $q$ elements and ${\mathbb{P}}^{n}(\mathbb{F}_{q})=\Pi_{n}$ the projective space of n-dimension over the field $\mathbb{F}_{q}$. Then

[TABLE]

Let $\mathcal{Q}$ be a quadric in $\mathbb{P}^{n}(\mathbb{F}_{q})$(i.e. $\mathcal{Q}=Z(F)$ where $F$ is a form of degree 2). The rank of $\mathcal{Q}$, denoted r$(\mathcal{Q})$, is the smallest number of indeterminates appearing in $F$ under any change of coordinate system. The quadric $\mathcal{Q}$ is said to be degenerate if r$(\mathcal{Q})<n+1$; otherwise it is non-degenerate. For $\mathcal{Q}$ a degenerate quadric and r$(\mathcal{Q})$=r, $\mathcal{Q}$ is a cone $\Pi_{n-r}\mathcal{Q}_{r-1}$ with vertex $\Pi_{n-r}$ (the set of the singular points of $\mathcal{Q}$) and base $\mathcal{Q}_{r-1}$ in a subspace $\Pi_{r-1}$ skew to $\Pi_{n-r}$.

Definition 2.1

For $\mathcal{Q}=\Pi_{n-r}\mathcal{Q}_{r-1}$ a degenerate quadric with r$(\mathcal{Q})=r$, $\mathcal{Q}_{r-1}$ is called the non-degenerate quadric associated to $\mathcal{Q}$. The degenerate quadric $\mathcal{Q}$ will be said to be of hyperbolic type (resp. elliptic, parabolic) if its associated non-degenerate quadric is of that type.

Definition 2.2

[13, p.158] Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two quadrics. The order $w(\mathcal{Q}_{1},\mathcal{Q}_{2})$ of the pair $(\mathcal{Q}_{1}$, $\mathcal{Q}_{2})$, is the minimum number of variables, after invertible linear change of variables, necessary to write $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$. When $w(\mathcal{Q}_{1},\mathcal{Q}_{2})=t+1$, we assume that $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are defined with the first $t+1$ indeterminates $x_{0},\ldots,x_{t}$ and we define $\mathbb{E}_{t}(\mathbb{F}_{q})=\{x\in\mathbb{P}^{n}(\mathbb{F}_{q})\ |x_{t+1}=...=x_{n}=0\}$.

Lemma 2.3

[5, pp.70-71 ] Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two quadrics in $\mathbb{P}^{n}(\mathbb{F}_{q})$ and $l$ an integer such that $1\leq l\leq n-1$. Suppose that $w(\mathcal{Q}_{1},\mathcal{Q}_{2})=n-l+1$ (i.e. there exists a linear transformation such that $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are defined with the indeterminates $x_{0},x_{1},...,x_{n-l}$) and $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}\cap{\mathbb{E}}_{n-l}(\mathbb{F}_{q})|\leq m$. Then

[TABLE]

This bound is optimal as soon as $m$ is optimal in $\mathbb{E}_{n-l}(\mathbb{F}_{q})$.

3 Resolution of the conjecture

Let us state the following result which will be very useful in the proof of the below theorem.

Lemma 3.1

For $n\geq 3$, $LS(n,q)\leq 4q^{n-2}+\pi_{n-3}$.

Proof: An easy computation.

Theorem 3.2

Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two quadrics in ${\mathbb{P}}^{n}(\mathbb{F}_{q})$ with no common hyperplane. Then

[TABLE]

And this bound is the best possible.

Proof: It has been proved in [6, 7] that the above conjecture is true for $n=3$ and 4. We will make an induction reasoning on the dimension of the projective space.

Let us suppose that the conjecture is true for $i$, $4\leq i\leq n-1$; that means for two quadrics $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ in ${\mathbb{P}}^{i}(\mathbb{F}_{q})$ with no common hyperplane we have

[TABLE]

Suppose now that $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are two quadrics in ${\mathbb{P}}^{n}(\mathbb{F}_{q})$ with no common hyperplane. Let us consider the order $w(\mathcal{Q}_{1},\mathcal{Q}_{2})$ of the two quadrics. We have either $w(\mathcal{Q}_{1},\mathcal{Q}_{2})=n+1$ or $w(\mathcal{Q}_{1},\mathcal{Q}_{2})\leq n$.

If $w(\mathcal{Q}_{1},\mathcal{Q}_{2})=n+1$, from the result of D. B. Leep and L. M. Schueller, we deduce that $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}|\leq LS(n,q)$. Therefore from the above Lemma we deduce that the conjecture is true.

If $w(\mathcal{Q}_{1},\mathcal{Q}_{2})\leq n$, then there is an integer $l$, $1\leq l\leq n-1$ such that $w(\mathcal{Q}_{1},\mathcal{Q}_{2})=n-l+1$. Therefore there exists a linear transformation such that $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are defined with the indeterminates $x_{0},x_{1},...,x_{n-l}$. And from the above hypothese we deduce that:

[TABLE]

Then by applying Lemma 2.3 with $m=4q^{n-l-2}+\pi_{n-l-3}$, we deduce that

[TABLE]

Let now $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two quadrics of ${\mathbb{P}}^{n}(\mathbb{F}_{q})$ defined by the equations $f_{1}(x_{0},x_{1},...,x_{n})=x_{0}^{2}+x_{1}^{2}-x_{2}^{2}$, and $f_{2}(x_{0},x_{1},...,x_{n})=x_{0}x_{1}$. In [Edoukou IEEE], it has been proved that (with $q$ odd), $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}\cap{\mathbb{E}}_{3}(\mathbb{F}_{q})|=4q+1$. Therefore by using the above Lemma, we deduce that, we have $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}|=4q^{n-2}+\pi_{n-3}$. If $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are defined the equations $f_{1}(x_{0},x_{1},...,x_{n})=(x_{0}+x_{1})x_{2}+x_{2}^{2}$, and $f_{2}(x_{0},x_{1},...,x_{n})=(x_{2}+x_{0})x_{1}+x_{1}^{2}$, we have $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}\cap{\mathbb{E}}_{2}(\mathbb{F}_{q})|=4$. And by the preceding reasoning we have $|\mathcal{Q}_{1}\cap\mathcal{Q}_{2}|=4q^{n-2}+\pi_{n-3}$.

Remark 3.3

It has been proved in [8] that when one of the two quadrics $\mathcal{Q}_{1}$ or $\mathcal{Q}_{2}$ is non-degenerate (hyperbolic, elliptic, parabolic), we have

[TABLE]

with

[TABLE]

These bounds are the best possible. Therefore, in this particular case, these bounds are better than the one of D. B. Leep and L. M. Schueller [13, p.172].

4 Conjectures on the number of points of algebraic sets

Here we will establish two conjectures concerning the number of points in the intersection of two degenerate quadrics with no common hyperplane and secondly for the number of points of a general algebraic set.

If $\mathcal{X}=\Pi_{n-1}\mathcal{P}_{0}$ is the degenerate quadric of rank r=1 (i.e. a repeated hyperplane) and $\mathcal{Q}$ any quadric not containing $\mathcal{X}$ we can easily prove that $|\mathcal{X}\cap\mathcal{Q}|\leq 2q^{n-2}+\pi_{n-3}$. And this upper bound is optimal if we take for example $\mathcal{X}$ and $\mathcal{Q}$ defined by the equations $f_{\mathcal{X}}(x_{0},x_{1},...,x_{n})={x_{2}}^{2}$, and $f_{\mathcal{Q}}(x_{0},x_{1},...,x_{n})=x_{0}x_{1}+{x_{2}}^{2}$.

If $\mathcal{X}=\Pi_{n-2}\mathcal{E}_{1}$ is the degenerate quadric of rank $r=2$ of elliptic type, for any quadric $\mathcal{Q}$ we have $|\mathcal{X}\cap\mathcal{Q}|\leq\pi_{n-2}$. And this upper bound is reached if we take for example $\mathcal{X}$ and $\mathcal{Q}$ defined by the equations $f_{\mathcal{X}}(x_{0},x_{1},...,x_{n})=f(x_{0},x_{1})$ ($f$ is irreducible), and $f_{\mathcal{Q}}(x_{0},x_{1},...,x_{n})=x_{0}x_{1}$.

We also know from the examples in the previous section that when $\mathcal{X}=\Pi_{n-r}\mathcal{Q}_{r-1}$ is a degenerate quadric of rank $r=2$ of hyperbolic type or rank $r=3$, we can find another degenerate quadric such that $|\mathcal{X}\cap\mathcal{Q}|=4q^{n-2}+\pi_{n-3}$.

Let us now suppose that $r\geq 4$. We have the following conjecture

Conjecture 4.1

Let $\mathcal{X}=\Pi_{n-r}\mathcal{Q}_{r-1}$ be a degenerate quadric of rank r$(\mathcal{X})=r$ and $\mathcal{Q}$ another degenerate quadric. If $r\geq 4$, then

[TABLE]

And this bound is the best possible

Let us remark that if the conjecture is true, that means for every functional code of order 2 defined on a degenerate quadric $\mathcal{X}=\Pi_{n-r}\mathcal{Q}_{r-1}$, we know exactly its minimum distance.

Based on the authors experiences on the study of the number of points in the intersection of hypersufaces, we would like to propose the following conjecture.

Conjecture 4.2

Let $X\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ be a projective algebraic set of degree $d$ and dimension $s$. Then,

[TABLE]

Remark 4.3

*Let us remark that this conjecture is true if X is of codimension 1. This is the Tsfasman-Serre-Sørensen’s upper bound for hypersurfaces [15], [16, chap.2, pp.7-10].

The Theorem 3.2 is a particular case of this conjecture. In fact, the intersection of the two quadrics with no common hyperplane is an algebraic set of dimension $n-2$ and degree $4$.*

Remark 4.4

There is also an upper bound concerning the number of points of a projective algebraic set which was already known by G. Lachaud since 1993 and proved by Lachaud[12, p.80], Boguslavsky[2, pp.288, 293-294] and Gorpade-Lachaud [9,pp.627-630]:

[TABLE]

In 1995, G. Lachaud [2, p. 292], [9, p.629] proposed the following conjecture on the upper bound for the number of points of a projective algebraic set:

[TABLE]

Our bound (1) is more better than the one of G. Lachaud (2). In [5, 6, 7, 8] we can find several cases where the Conjecture 4.2 is true and where the upper bound of G. Lachaud (2) seems to be too large.

**References

**[1] Y. Aubry, Reed-Muller codes associated to projective varieties, in ”Coding Theory and Algebraic Geometry, Luminy, 1991,” Lecture Notes in Math. Vol. 1518 Springer-Verlag, Berlin, 4-17, 1992.

[2] M. Boguslavsky, On the number of points of polynomial systems. Finite Fields and Theirs Applications 3, (1999), 287-299.

[3] A. A. Bruen and J. W. P. Hirschfeld, Intersections in projective space I: Combinatorics. Mathematische Zeitschrift, Vol. 193, (1986), 215-226.

[4] L. Carlitz, Pairs of quadratic equations in a finite field. America Journal of Mathematics, Vol. 76, 137-154, (1954).

[5 ] F. A. B. Edoukou, Codes correcteurs d’erreurs construits à partir des variétés algébriques. Ph.D Thesis, Université de la Méditerranée (Aix-Marseille II), France, (2007).

[6] F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric surfaces. IEEE Trans. Inf. Vol 54 N0.(2), 860-864, (2008).

[7] F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric varieties in $\mathbb{P}^{4}(\mathbb{F}_{q})$. Contemporary Mathematics, Proceedings of the 11 th Conference AGC2T, Marseille, France, November 2007, 21-32, (2009).

[8] F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes $C_{2}(\mathcal{Q})$, $\mathcal{Q}$ a non-singular quadric. 19 pages, Submitted to Journal of Pure and Applied Algebra (2009).

[9] S. R. Ghorpade and G. Lachaud, Etale cohomology, Lefstchetz theorems and number of points of singular varieties over finite fields. Moscow Mathematical Journal, Vol. 2, N0.3, (2002), 589-631.

[10] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields (Second Edition) Clarendon Press. Oxford 1998.

[11] J. W. P. Hirschfeld, General Galois Geometries, Clarendon press. Oxford 1991.

[12] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties; in ” Arithmetic, Geometry, and Coding Theory ”. (Luminy, France, 1993), Walter de Gruyter, Berlin-New York, (1996), 77-104.

[13] D. B. Leep and L. M. Schueller, Zeros of a pair of quadric forms defined over finite field. Finite Fields and Their Applications 5 (1999), 157-176.

[14] W. M. Schmidt, Equations over finite fields. An Elementary Approach, Lecture Notes in Maths 536 (1975).

[15] J. -P. Serre, Lettre à M. Tsfasman, In ”Journées Arithmétiques de Luminy (1989)”, Astérisque 198-199-200 (1991), 3511-353.

[16] A. B. Sørensen, Rational points on hypersurfaces, Reed-Muller codes and algebraic-geometric codes. Ph. D. Thesis, Aarhus, Denmark, 1991.

[17] A. Weil, Footnote to a recent paper. American Journal of Mathematics, Vol. 76, 347-350, (1954).