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

ON THE (WEAK) CYLINDER CONJECTURE Finite Geometry Workshop Szeged 2017 April 29, 2017 jan@debeule.eu 1

THREE MUSKETEERS IN PÉCS, 2016 2

Introduction THE STATEMENT Conjecture (S. Ball 2008) Let q be prime. Let U be a set of q 2 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. 3

Introduction THE STATEMENT Conjecture (S. Ball 2008) Let q be prime. Let U be a set of q 2 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. 3

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 . 4

Combinatorics COMBINATORIAL OBSERVATIONS Definition Let π be a plane of AG ( 3 , q ) . Let n π := | π ∩ U | and when | π ∩ U | � = 0, call n π q − 1 the excess of π . 5

Combinatorics COMBINATORIAL OBSERVATIONS Definition Let π be a plane of AG ( 3 , q ) . Let n π := | π ∩ U | and when | π ∩ U | � = 0, call n π 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. 5

Combinatorics COMBINATORIAL OBSERVATIONS Definition Let π be a plane of AG ( 3 , q ) . Let n π := | π ∩ U | and when | π ∩ U | � = 0, call n π 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 . 5

Combinatorics COMBINATORIAL OBSERVATIONS Lemma The cylinder conjecture is true for q = 3 . 6

Combinatorics COMBINATORIAL OBSERVATIONS Lemma The cylinder conjecture is true for q = 3 . Conjecture The cylinder conjecture is true for q = 5 . 6

The weak cylinder conjecture STATEMENT Conjecture (S. Ball 2008) Let q be prime. Let U be a set of q 2 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. 7

The weak cylinder conjecture INTERSECTION NUMBERS Lemma Let q be prime. Let U be a set of q 2 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 ) . 8

The weak cylinder conjecture INTERSECTION NUMBERS U = { ( a i , b i , c i , 1 ) | i = 1 , . . . , q 2 } . 9

The weak cylinder conjecture INTERSECTION NUMBERS U = { ( a i , b i , c i , 1 ) | i = 1 , . . . , q 2 } . π ∞ : W = 0 9

The weak cylinder conjecture INTERSECTION NUMBERS U = { ( a i , b i , c i , 1 ) | i = 1 , . . . , q 2 } . π ∞ : W = 0 π [ x , z , y , w ] : xX + yY + zZ + wW = 0 9

The weak cylinder conjecture INTERSECTION NUMBERS U = { ( a i , b i , c i , 1 ) | i = 1 , . . . , q 2 } . π ∞ : W = 0 π [ x , z , y , w ] : xX + yY + zZ + wW = 0 l [ x , y , z ] : xX + yY + zZ = W = 0 9

The weak cylinder conjecture INTERSECTION NUMBERS q 2 � R ( X , Y , Z , W ) := ( a i X + b i Y + c i Z + W ) . i = 1 10

The weak cylinder conjecture INTERSECTION NUMBERS q 2 � R ( X , Y , Z , W ) := ( a i X + b i Y + c i Z + W ) . i = 1 R ( x , y , z , w ) = 0 ⇐ ⇒ π [ x , y , z , w ] contains ( a i , b i , c i , 1 ) . 10

The weak cylinder conjecture INTERSECTION NUMBERS q 2 � R ( X , Y , Z , W ) := ( a i X + b i Y + c i Z + W ) . i = 1 R ( x , y , z , w ) = 0 ⇐ ⇒ π [ x , y , z , w ] contains ( a i , b i , c i , 1 ) . q 2 R ( X , Y , Z , W ) = W q 2 + σ j ( X , Y , Z ) W q 2 − j . � j = 1 10

The weak cylinder conjecture INTERSECTION NUMBERS q 2 R ( X , Y , Z , W ) = W q 2 + σ j ( X , Y , Z ) W q 2 − j . � j = 1 11

The weak cylinder conjecture INTERSECTION NUMBERS q 2 R ( X , Y , Z , W ) = W q 2 + σ j ( X , Y , Z ) W q 2 − j . � j = 1 q 2 ( a i X + b i Y + c i Z ) j , � S j ( X , Y , Z ) := i = 1 k � ( − 1 ) i − 1 S i ( X , Y , Z ) σ k − i ( X , Y , Z ) . k σ k ( X , Y , Z ) = i = 1 11

The weak cylinder conjecture INTERSECTION NUMBERS Lemma The polynomials σ i ( X , Y , Z ) = 0 = S i ( X , Y , Z ) , i = 1 . . . q − 1 . 12

The weak cylinder conjecture INTERSECTION NUMBERS Lemma The polynomials σ i ( X , Y , Z ) = 0 = S i ( X , Y , Z ) , i = 1 . . . q − 1 . q 2 � ( a i X + b i Y + c i Z + W ) q − 1 G ( X , Y , Z , W ) := i = 1 12

The weak cylinder conjecture INTERSECTION NUMBERS Lemma The polynomials σ i ( X , Y , Z ) = 0 = S i ( X , Y , Z ) , i = 1 . . . q − 1 . q 2 � ( a i X + b i Y + c i Z + W ) q − 1 G ( X , Y , Z , W ) := i = 1 q 2 q − 1 � p − 1 � � � ( a i X + b i Y + c i Z ) j W p − 1 − j . G ( X , Y , Z , W ) = j i = 1 j = 0 12

The weak cylinder conjecture INTERSECTION NUMBERS Lemma The polynomials σ i ( X , Y , Z ) = 0 = S i ( X , Y , Z ) , i = 1 . . . q − 1 . q 2 � ( a i X + b i Y + c i Z + W ) q − 1 G ( X , Y , Z , W ) := i = 1 q 2 q − 1 � p − 1 � � � ( a i X + b i Y + c i Z ) j W p − 1 − j . G ( X , Y , Z , W ) = j i = 1 j = 0 p − 1 � p − 1 � S j ( X , Y , Z ) W p − 1 − j . � G ( X , Y , Z , W ) = j j = 0 12

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. 13

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 . 13

The weak cylinder conjecture THE WEAKER CYLINDER CONJECTURE We assume that at least q + 1 directions are not determined, then σ q ( X , Y , Z ) = 0. 14

The weak cylinder conjecture AN EXAMPLE q + 1 2 | x ∈ F q } , this set determines q + 3 q odd, S = { ( x , x points. 2 15

Some nice polynomials SOME NICE POLYNOMIALS Lemma The polynomials σ i ( X , Y , Z ) = 0 , i = 1 . . . q . 16

Some nice polynomials SOME NICE POLYNOMIALS Lemma The polynomials σ i ( 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 . 16

Some nice polynomials SOME NICE POLYNOMIALS Lemma The polynomials σ i ( 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 16

Some nice polynomials THE WEAKER CYLINDER CONJECTURE Lemma Under the assumptions for the set U , the following polynomial identities hold. σ k ( Y , Z , W ) ≡ 0 , k = lq + 1 . . . ( l + 1 ) q − l , l = 0 . . . q − 1 , ( − j + 1 ) σ j + q − 1 ( Y , Z , W ) + ( Y q ∂σ j ∂ Y + Z q ∂σ j ∂ Z + W q ∂σ j ∂ W ) ≡ 0 , j = q + 1 . . . q 2 − q , Y q ∂σ j ∂ Y + Z q ∂σ j ∂ Z + W q ∂σ j ∂ W ≡ 0 , j = q 2 − q + 1 . . . q 2 . 17

Some nice polynomials INTERSECTIONS WITH LINES ◮ Consider the planes π s := π [ s , 1 , 0 , α ] and π t := π [ t , 0 , 1 , − β ] ◮ Define l s , t := π s ∩ π t , this is a line through ( 1 , − s , − t , 0 ) . ◮ R s , t ( Y , Z , W ) := R ( sY + tZ , Y , Z , W ) “describes” the intersection of U with lines. 18

Some nice polynomials INTERSECTIONS WITH LINES ◮ Consider the planes π s := π [ s , 1 , 0 , α ] and π t := π [ t , 0 , 1 , − β ] ◮ Define l s , t := π s ∩ π t , this is a line through ( 1 , − s , − t , 0 ) . ◮ R s , t ( Y , Z , W ) := R ( sY + tZ , Y , Z , W ) “describes” the intersection of U with lines. G s , t ( Y , Z , W ) := G ( sY + tZ , Y , Z , W ) idea: try to prove that for a fixed ( s , t ) the polynomial R s , t ( Y , Z , W ) is a q th power. 18

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 ′ = { ( a i , b i ) | i = 1 . . . q 2 } Define w ( x , y ) : F q × F q → N as the number of times that ( x , y ) ∈ U ′ . 19

A slightly different approach PROPERTIES OF THE WEIGHT FUNCTION ◮ w ( x , 0 ) = 0 for all x ∈ F q , w ( 0 , y ) = 0 for all y ∈ F q . ◮ Let a , b ∈ F q , then � x ∈ F q w ( x , ax + b ) ≡ 0 ( mod q ) . x , y ∈ F q w ( x , y ) ≤ q 2 . ◮ � 20