some contributions to incidence geometry and the polynomial … · 2016. 12. 21. · 1 incidence...

Some Contributions to Incidence Geometry and thePolynomial Method

Anurag Bishnoi

Department of MathematicsGhent University

https://anuragbishnoi.wordpress.com/

Promoter: Prof. Dr. Bart De Bruyn

https://anuragbishnoi.wordpress.com/

Outline

1 Incidence GeometryIntroductionNew Near PolygonsSemi-Finite Generalized PolygonsCharacterization of Suzuki Tower Near Polygons

2 Polynomial MethodIntroductionAlon-FürediPunctured Chevalley-Warning

Anurag Bishnoi thesis 2 / 28

INCIDENCE GEOMETRY

Introduction

What is left of the usual Euclidean geometry when you remove the notionsof distances, angles, betweenness, etc. and only keep the notion of“incidences” between points and lines.

A B

CD

`

m

n

r

Points: A, B, C, D Lines: `, m, n, r

the point A is incident with the lines ` and m

Introduction

What is left of the usual Euclidean geometry when you remove the notionsof distances, angles, betweenness, etc. and only keep the notion of“incidences” between points and lines.

A B

CD

`

m

n

r

Points: A, B, C, D Lines: `, m, n, r

the point A is incident with the lines ` and m

The Fano plane

A C

E

G

B

F D

7 points and 7 lines, order (2, 2)

The Fano plane

A C

E

G

B

F D

The Fano plane

A C

E

G

B

F D

The Doily

Automorphism Group

Most of the geometries that we study are highly “symmetrical”.Mathematically, these symmetries are certainrearrangements/permutations of points which map every line to anotherline.

For example, in the square example, look at the permutation A 7→ B,B 7→ C, C 7→ D, D 7→ A:

A B

CD

`

m

n

r

D A

BC

r

`

m

n

The 4-gon has in total 8 automorphisms (symmetries), the Fano plane has168 and the Doily has 720.

Automorphism Group

Most of the geometries that we study are highly “symmetrical”.Mathematically, these symmetries are certainrearrangements/permutations of points which map every line to anotherline. For example, in the square example, look at the permutation A 7→ B,B 7→ C, C 7→ D, D 7→ A:

A B

CD

`

m

n

r

D A

BC

r

`

m

n

Automorphism Group

Most of the geometries that we study are highly “symmetrical”.Mathematically, these symmetries are certainrearrangements/permutations of points which map every line to anotherline. For example, in the square example, look at the permutation A 7→ B,B 7→ C, C 7→ D, D 7→ A:

A B

CD

`

m

n

r

D A

BC

r

`

m

n

Near Polygons and Generalized Polygons

Near polygons and generalized polygons are two important classes ofincidence geometries, which are the central objects of my PhD thesis.

Mathematical Definition:A near 2d-gon is a point-line geometry (P,L, I) which satisfies thefollowing axioms:

the collinearity graph is connected with finite diameter d ;

for every point x ∈ P and every line L ∈ L, there exists a uniquepoint incidence with L which is nearest to x .

A generalized n-gon is a point-line geometry whose incidence graph hasdiameter n and the maximum possible girth, 2n. For n even, everygeneralized n-gon is a near n-gon.

Near Polygons and Generalized Polygons

Near polygons and generalized polygons are two important classes ofincidence geometries, which are the central objects of my PhD thesis.

Mathematical Definition:A near 2d-gon is a point-line geometry (P,L, I) which satisfies thefollowing axioms:

the collinearity graph is connected with finite diameter d ;

for every point x ∈ P and every line L ∈ L, there exists a uniquepoint incidence with L which is nearest to x .

A generalized n-gon is a point-line geometry whose incidence graph hasdiameter n and the maximum possible girth, 2n. For n even, everygeneralized n-gon is a near n-gon.

The Doily

`C B

A

Doily: both a near quadrangle and a generalized quadrangle

The G2(4)-near octagon

A new incidence geometry of order (2, 10) that has 4,095 points, 15,015lines and an automorphism group of size 503,193,600.

For mathematicians: it’s a near octagon of order (2, 10) whoseautomorphism group is isomorphic to a split extension of the finite simplegroup G2(4) by C2, and it has connections with the Suzuki tower of finitesimple groups,

L3(2) < U3(3) < J2 < G2(4) < Suz .

The G2(4)-near octagon

A new incidence geometry of order (2, 10) that has 4,095 points, 15,015lines and an automorphism group of size 503,193,600.

For mathematicians: it’s a near octagon of order (2, 10) whoseautomorphism group is isomorphic to a split extension of the finite simplegroup G2(4) by C2, and it has connections with the Suzuki tower of finitesimple groups,

L3(2) < U3(3) < J2 < G2(4) < Suz .

The L3(4)-near octagon

Another new incidence geometry. It has order (2, 4), 315 points, 525 linesand an automorphism group of size 80640.

For the mathematicians: it’s a near octagon of order (2, 4) whoseautomorphism group is isomorphic to a split extension of the finite simplegroup PSL3(4) by C2 × C2 and its points are the nontrivial elations ofPG(2, 4)

Historical Context

Near polygons introduced in 1980 by Ernie Shult and ArthurYanushka for studying certain sets of lines in the Euclidean space.

Several families of these geometries were then constructed in the 80s,and they were found to have connections with other objects like polarspaces, distance regular graphs and sporadic finite simple groups.

Last new “nice” near polygon were discovered 15 years ago by BartDe Bruyn as a part of an infinite family.

Historical Context

Semi-Finite Generalized Polygons

Tits asked the following question: are there generalized polygons whichhave finitely many points (≥ 3) on each line but infinitely many linesthrough each point?

Very little progress has been made over the years, as only the case ofgeneralized quadrangles with 3, 4 or 5 points on each line is solved.

Semi-finite Generalized Hexagons

We answer a modified version of the question for the case of generalizedhexagons.

Theorem

Let q ∈ {2, 3, 4} and let S be a generalized hexagon isomorphic to thesplit Cayley hexagon H(q) or its dual H(q)D . Then the following holds forany generalized hexagon S ′ that contains S as a full subgeometry:(1) S ′ is finite;(2) if q ∈ {2, 4} and S ∼= H(q), then S ′ = S.

Valuation Theory of Near Polygons

A CB

IF

DG

H E

A CB

DEF

IHG

A 3× 3 grid inside a Doily

Some Characterizations

Theorem

The dual twisted triality hexagon T(2, 8) is the unique near hexagon oforder (2, 8) which contains H(2)D as a subgeometry.

Theorem

The Hall-Janko near octagon is the unique near octagon of order (2, 4)which contains the dual split Cayley hexagon H(2)D as an isometricallyembedded subgeometry.

Theorem

The G2(4)-near octagon is the unique near octagon of order (2, 10) whichcontains the Hall-Janko near octagon as an isometrically embeddedsubgeometry.

POLYNOMIAL METHOD

Covering all but one

Given finite sets of real numbers A and B, a finite grid is the set A× B ⊂ R2 ofpoints with coordinates (x , y) where x ∈ A and y ∈ B

A polynomial method approach

Each line in the plane is given by a linear equation of the formax + by − c = 0, i.e., it is the set of zeros of the polynomialf (x , y) = ax + by − c . So, the set of points covered by k lines`1, `2, . . . , `k is equal to the zero set of the degree k polynomialf = f1f2 · · · fk , where f1, . . . , fk are the linear polynomials that define thelines.

Lemma

Given a finite grid A× B, if polynomial f (x , y) vanishes on all points ofthe grid except one then the degree of f is at least |A| − 1 + |B| − 1.

For mathematicians: of course this also generalizes to n variablepolynomials and n-dimensional finite grids A1 × · · · × An.

Lemma

Given a finite grid A× B, if polynomial f (x , y) vanishes on all points ofthe grid except one then the degree of f is at least |A| − 1 + |B| − 1.

For mathematicians: of course this also generalizes to n variablepolynomials and n-dimensional finite grids A1 × · · · × An.

A more general problem

Given k hyperplanes and a finite grid A which is not completely covered bythe hyperplanes, how many points do the hyperplanes miss?

Let f (x1, . . . , xn) be an n-variable polynomial of degree d such that thereis at least some point (a1, . . . , an) of A where f (a1, . . . , an) 6= 0. Find thenumber of points of A where f does not vanish.

Or at least give a lower bound on the number of such points of A as afunction of d , |A1|, . . . , |An|.

Balls in Bins

A1

A2

An

Bin Ai holds at most |Ai | balls.

Balls in Bins

A1

A2

An

Bin Ai holds at most |Ai | balls. Distribution of k balls is an n-tupley = (y1, . . . , yn) with y1 + . . . + yn = k and 1 ≤ yi ≤ |Ai | for all i .

Balls in Bins

A1

A2

An

Let Π(y) = y1 · · · yn. If n ≤ k ≤ |A1|+ . . . + |An|, let m(|A1|, . . . , |An|; k) be theminimum value of Π(y) as y ranges over all distributions of k balls into binsA1, . . . ,An.

Alon-Füredi theorem

Theorem (Alon-Füredi)

Given a finite grid A = A1 × · · · × An and a polynomial f of degree dwhich does not vanish on all points of A, there are at leastm(|A1|, . . . , |An|;

∑|Ai | − d) points of A where f does not vanish.

Clearly if d <∑

(|Ai | − 1), then we have at least n + 1 balls, and hencethe minimum is at least 2. Therefore, the minimum number ofhyperplanes required to cover all points of the grid A except one is equalto

∑(|Ai | − 1).

Alon-Füredi theorem

Theorem (Alon-Füredi)

Given a finite grid A = A1 × · · · × An and a polynomial f of degree dwhich does not vanish on all points of A, there are at leastm(|A1|, . . . , |An|;

∑|Ai | − d) points of A where f does not vanish.

Clearly if d <∑

(|Ai | − 1), then we have at least n + 1 balls, and hencethe minimum is at least 2. Therefore, the minimum number ofhyperplanes required to cover all points of the grid A except one is equalto

∑(|Ai | − 1).

Applications

Coding Theory: Reed-Muller codes and their generalisations

Polynomial Identity Testing: Schwartz-Zippel lemma

Finite Geometry: blocking sets, hyperplane coverings

Number Theory: Chevalley-Warning theorems, zero sum problems

Graph Theory: (Alon-Friedland-Kalai) every 4-regular graph plus anedge contains a 3-regular subgraph

Generalized Alon-Füredi

Theorem (Generalized Alon-Füredi Theorem)

Let R be a ring and let A1, . . . ,An be nonempty finite subsets of R thatsatisfy Condition (D)a. For i ∈ {1, . . . , n}, let ai = |Ai | and let bi be aninteger such that 1 ≤ bi ≤ ai . Let f ∈ R[t1, . . . , tn] be a non-zeropolynomial such that degti f ≤ ai − bi for all i ∈ {1, . . . , n}. LetUA = {x ∈ A | f (x) 6= 0} where A = A1 × · · · × An ⊆ Rn. Then we have

|UA| ≥ m(a1, . . . , an; b1, . . . bn;n∑

i=1

ai − deg f ).

Moreover, for any such R, A1 . . . ,An and integers b1, . . . , bn, we canconstruct a polynomial f which meets this bound.

aFor any two distinct elements a, b of the subset, a− b is not a zero divisor.

Chevalley-Warning Theorem

Recall that

Lemma

A polynomial f which vanishes on all points of a finite grid A1 × · · · × Anexcept one, must have degree at least

∑ni=1(|Ai | − 1).

From this lemma, a classical result in mathematics called theChevalley-Warning theorem directly follows, which is a number theoreticresult that has found several applications in combinatorics.

Punctured Chevalley-Warning Theorem

Lemma (Ball and Serra)

A polynomial f which vanishes on all points of a finite grid A1 × · · · × Anexcept at some point of a subgrid B1× · · · ×Bn, must have degree at least∑n

i=1(|Ai | − |Bi |).

Theorem (Punctured Chevalley-Warning)

Let f1, . . . , fr ∈ Fq[t1, . . . , tn] be such that

(q − 1)r∑

j=1

deg fj <n∑

i=1

(|Ai | − |Bi |)

and let ZA be the set of common zeros of the fj ’s in the grid A. IfZA ∩ B 6= ∅, then ZA ∩ (A \ B) 6= ∅.

Thank you!

Incidence GeometryIntroductionNew Near PolygonsSemi-Finite Generalized PolygonsCharacterization of Suzuki Tower Near Polygons

Polynomial MethodIntroductionAlon-FürediPunctured Chevalley-Warning

some contributions to incidence geometry and the polynomial … · 2016. 12. 21. · 1 incidence...

Documents