ON THE (WEAK) CYLINDER CONJECTURE Finite Geometry Workshop Szeged - - PowerPoint PPT Presentation

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ON THE (WEAK) CYLINDER CONJECTURE

Finite Geometry Workshop Szeged 2017 April 29, 2017

jan@debeule.eu

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THREE MUSKETEERS IN PÉCS, 2016

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Introduction

THE STATEMENT Conjecture (S. Ball 2008)

Let q be prime. Let U be a set of q2 points of AG(3, q) such that for every hyperplane π of AG(3, q) |U ∩ π| ≡ 0 (mod q) . Then U is the set of points of q parallel lines.

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Introduction

THE STATEMENT Conjecture (S. Ball 2008)

Let q be prime. Let U be a set of q2 points of AG(3, q) such that for every hyperplane π of AG(3, q) |U ∩ π| ≡ 0 (mod q) . Then U is the set of points of q parallel lines.

Definition

A cylinder in AG(3, q) is the set of points of q parallel lines.

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Combinatorics

RICH AND EMPTY PLANES

Let U be a set of points of AG(3, q) satisfying the conditions of the cylinder conjecture.

Definition

Call a plane rich if it contains more than q points of U. Call a plane empty if it contains no points of U.

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Combinatorics

COMBINATORIAL OBSERVATIONS Definition

Let π be a plane of AG(3, q). Let nπ := |π ∩ U| and when |π ∩ U| = 0, call nπ

q − 1 the excess of π.

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Combinatorics

COMBINATORIAL OBSERVATIONS Definition

Let π be a plane of AG(3, q). Let nπ := |π ∩ U| and when |π ∩ U| = 0, call nπ

q − 1 the excess of π.

Lemma

Let l be a line meeting U in k > 0 points. Then the sum of the excess of the q + 1 planes on l equals k − 1.

Corollary

There exists rich planes and empty planes.

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Combinatorics

COMBINATORIAL OBSERVATIONS Definition

Let π be a plane of AG(3, q). Let nπ := |π ∩ U| and when |π ∩ U| = 0, call nπ

q − 1 the excess of π.

Lemma

Let l be a line meeting U in k > 0 points. Then the sum of the excess of the q + 1 planes on l equals k − 1.

Corollary

There exists rich planes and empty planes.

Corollary

If there is only one rich plane π, then U is the set of points of π.

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Combinatorics

COMBINATORIAL OBSERVATIONS Lemma

The cylinder conjecture is true for q = 3.

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Combinatorics

COMBINATORIAL OBSERVATIONS Lemma

The cylinder conjecture is true for q = 3.

Conjecture

The cylinder conjecture is true for q = 5.

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The weak cylinder conjecture

STATEMENT Conjecture (S. Ball 2008)

Let q be prime. Let U be a set of q2 points of AG(3, q) and let N be the set of non-determined directions. If |N| ≥ p, then U is the set of points of a cylinder.

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The weak cylinder conjecture

INTERSECTION NUMBERS Lemma

Let q be prime. Let U be a set of q2 points of AG(3, q) and let N be the set of non-determined directions. If |N| ≥ q, then for every plane π of AG(2, q) |π ∩ U| ≡ 0 (mod q) .

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The weak cylinder conjecture

INTERSECTION NUMBERS

U = {(ai, bi, ci, 1)|i = 1, . . . , q2}.

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The weak cylinder conjecture

INTERSECTION NUMBERS

U = {(ai, bi, ci, 1)|i = 1, . . . , q2}. π∞ : W = 0

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The weak cylinder conjecture

INTERSECTION NUMBERS

U = {(ai, bi, ci, 1)|i = 1, . . . , q2}. π∞ : W = 0 π[x, z, y, w] : xX + yY + zZ + wW = 0

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The weak cylinder conjecture

INTERSECTION NUMBERS

U = {(ai, bi, ci, 1)|i = 1, . . . , q2}. π∞ : W = 0 π[x, z, y, w] : xX + yY + zZ + wW = 0 l[x, y, z] : xX + yY + zZ = W = 0

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The weak cylinder conjecture

INTERSECTION NUMBERS

R(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W ).

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The weak cylinder conjecture

INTERSECTION NUMBERS

R(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W ). R(x, y, z, w) = 0 ⇐ ⇒ π[x, y, z, w] contains (ai, bi, ci, 1).

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The weak cylinder conjecture

INTERSECTION NUMBERS

R(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W ). R(x, y, z, w) = 0 ⇐ ⇒ π[x, y, z, w] contains (ai, bi, ci, 1). R(X, Y , Z, W ) = W q2 +

q2

• j=1

σj(X, Y , Z)W q2−j .

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The weak cylinder conjecture

INTERSECTION NUMBERS

R(X, Y , Z, W ) = W q2 +

q2

• j=1

σj(X, Y , Z)W q2−j .

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The weak cylinder conjecture

INTERSECTION NUMBERS

R(X, Y , Z, W ) = W q2 +

q2

• j=1

σj(X, Y , Z)W q2−j . Sj(X, Y , Z) :=

q2

• i=1

(aiX + biY + ciZ)j , kσk(X, Y , Z) =

k

• i=1

(−1)i−1Si(X, Y , Z)σk−i(X, Y , Z).

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The weak cylinder conjecture

INTERSECTION NUMBERS Lemma

The polynomials σi(X, Y , Z) = 0 = Si(X, Y , Z), i = 1 . . . q − 1.

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The weak cylinder conjecture

INTERSECTION NUMBERS Lemma

The polynomials σi(X, Y , Z) = 0 = Si(X, Y , Z), i = 1 . . . q − 1. G(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W )q−1

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The weak cylinder conjecture

INTERSECTION NUMBERS Lemma

The polynomials σi(X, Y , Z) = 0 = Si(X, Y , Z), i = 1 . . . q − 1. G(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W )q−1 G(X, Y , Z, W ) =

q2

• i=1

q−1

• j=0

p − 1 j

• (aiX + biY + ciZ)jW p−1−j.

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The weak cylinder conjecture

INTERSECTION NUMBERS Lemma

The polynomials σi(X, Y , Z) = 0 = Si(X, Y , Z), i = 1 . . . q − 1. G(X, Y , Z, W ) :=

q2

• i=1

(aiX + biY + ciZ + W )q−1 G(X, Y , Z, W ) =

q2

• i=1

q−1

• j=0

p − 1 j

• (aiX + biY + ciZ)jW p−1−j.

G(X, Y , Z, W ) =

p−1

• j=0

p − 1 j

• Sj(X, Y , Z)W p−1−j .

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The weak cylinder conjecture

BACK TO THE WEAK CYLINDER CONJECTURE Corollary

Every plane π[x, y, z, w] meets U in 0 (mod q) points.

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The weak cylinder conjecture

BACK TO THE WEAK CYLINDER CONJECTURE Corollary

Every plane π[x, y, z, w] meets U in 0 (mod q) points.

Theorem (Ball, Govaerts, Storme 2006)

If the set N of non-determined directions contains a conic, then U is the set of points of a plane not meeting N.

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The weak cylinder conjecture

THE WEAKER CYLINDER CONJECTURE

We assume that at least q + 1 directions are not determined, then σq(X, Y , Z) = 0.

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The weak cylinder conjecture

AN EXAMPLE

q odd, S = {(x, x

q+1 2 |x ∈ Fq}, this set determines q+3

2

points.

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Some nice polynomials

SOME NICE POLYNOMIALS Lemma

The polynomials σi(X, Y , Z) = 0, i = 1 . . . q.

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Some nice polynomials

SOME NICE POLYNOMIALS Lemma

The polynomials σi(X, Y , Z) = 0, i = 1 . . . q.

Lemma

R · (G − d) = (X q − X) ∂R

∂X + (Y q − Y ) ∂R ∂Y + (Z q − Z) ∂R ∂Z + (W q − W ) ∂R ∂W .

Lemma

d · R = X ∂R

∂X + Y ∂R ∂Y + Z ∂R ∂Z + W ∂R ∂W .

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Some nice polynomials

SOME NICE POLYNOMIALS Lemma

The polynomials σi(X, Y , Z) = 0, i = 1 . . . q.

Lemma

R · (G − d) = (X q − X) ∂R

∂X + (Y q − Y ) ∂R ∂Y + (Z q − Z) ∂R ∂Z + (W q − W ) ∂R ∂W .

Lemma

d · R = X ∂R

∂X + Y ∂R ∂Y + Z ∂R ∂Z + W ∂R ∂W .

Corollary

G · R = X q ∂R

∂X + Y q ∂R ∂Y + Z q ∂R ∂Z + W q ∂R ∂W

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Some nice polynomials

THE WEAKER CYLINDER CONJECTURE Lemma

Under the assumptions for the set U, the following polynomial identities hold. σk(Y , Z, W ) ≡ 0 , k = lq + 1 . . . (l + 1)q − l, l = 0 . . . q − 1 , (−j + 1)σj+q−1(Y , Z, W ) + (Y q ∂σj ∂Y + Z q ∂σj ∂Z + W q ∂σj ∂W ) ≡ 0 , j = q + 1 . . . q2 − q , Y q ∂σj ∂Y + Z q ∂σj ∂Z + W q ∂σj ∂W ≡ 0 , j = q2 − q + 1 . . . q2 .

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Some nice polynomials

INTERSECTIONS WITH LINES

◮ Consider the planes πs := π[s, 1, 0, α] and πt := π[t, 0, 1, −β] ◮ Define ls,t := πs ∩ πt, this is a line through (1, −s, −t, 0). ◮ Rs,t(Y , Z, W ) := R(sY + tZ, Y , Z, W ) “describes” the

intersection of U with lines.

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Some nice polynomials

INTERSECTIONS WITH LINES

◮ Consider the planes πs := π[s, 1, 0, α] and πt := π[t, 0, 1, −β] ◮ Define ls,t := πs ∩ πt, this is a line through (1, −s, −t, 0). ◮ Rs,t(Y , Z, W ) := R(sY + tZ, Y , Z, W ) “describes” the

intersection of U with lines. Gs,t(Y , Z, W ) := G(sY + tZ, Y , Z, W ) idea: try to prove that for a fixed (s, t) the polynomial Rs,t(Y , Z, W ) is a qth power.

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A slightly different approach

PROJECTING FROM APEX

◮ Empty planes: π[1, 0, 0, 0] and π[0, 1, 0, 0], intersection point

at infinity: apex a = (0, 0, 1, 0).

◮ Projection from a on a plane not through a.

multiset U′ = {(ai, bi)|i = 1 . . . q2} Define w(x, y) : Fq × Fq → N as the number of times that (x, y) ∈ U′.

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A slightly different approach

PROPERTIES OF THE WEIGHT FUNCTION

◮ w(x, 0) = 0 for all x ∈ Fq, w(0, y) = 0 for all y ∈ Fq. ◮ Let a, b ∈ Fq, then x∈Fq w(x, ax + b) ≡ 0 (mod q). ◮ x,y∈Fq w(x, y) ≤ q2.

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A slightly different approach

PROPERTIES OF THE WEIGHT FUNCTION

◮ w(x, 0) = 0 for all x ∈ Fq, w(0, y) = 0 for all y ∈ Fq. ◮ Let a, b ∈ Fq, then x∈Fq w(x, ax + b) ≡ 0 (mod q). ◮ x,y∈Fq w(x, y) ≤ q2.

If a full line on a is contained in U, we can delete it, properties above remain unchanged, and furthermore

◮ w(x, y) < p for all x, y ∈ Fq

Lemma

• x,y∈Fq w(x, y)xkyl = 0, for all k, l such that k + l ≤ q.

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A slightly different approach

A COMPUTATIONAL APPROACH Lemma

There exists no polynomial w(X, Y ) as described for q ≤ 13.

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A slightly different approach

BIBLIOGRAPHY

Simeon Ball and Michel Lavrauw. How to use Rédei polynomials in higher dimensional spaces. Matematiche (Catania), 59(1-2):39–52 (2006), 2004. Simeon Ball and Michel Lavrauw. On the graph of a function in two variables over a finite field.

• J. Algebraic Combin., 23(3):243–253, 2006.

Simeon Ball. On the graph of a function in many variables over a finite field.

• Des. Codes Cryptogr., 47(1-3):159–164, 2008.