Some Contributions to Incidence Geometry and the Polynomial Method

Transcription

1 Some Contributions to Incidence Geometry and the Polynomial Method Anurag Bishnoi Department of Mathematics Ghent University Promoter: Prof. Dr. Bart De Bruyn

2 Outline 1 Incidence Geometry Introduction New Near Polygons Semi-Finite Generalized Polygons Characterization of Suzuki Tower Near Polygons 2 Polynomial Method Introduction Alon-Füredi Punctured Chevalley-Warning Anurag Bishnoi thesis 2 / 28

3 INCIDENCE GEOMETRY Anurag Bishnoi thesis 3 / 28

4 Introduction What is left of the usual Euclidean geometry when you remove the notions of distances, angles, betweenness, etc. and only keep the notion of incidences between points and lines. Anurag Bishnoi thesis 4 / 28

5 Introduction What is left of the usual Euclidean geometry when you remove the notions of distances, angles, betweenness, etc. and only keep the notion of incidences between points and lines. D r C l n A m B Points: A, B, C, D Lines: l, m, n, r the point A is incident with the lines l and m Anurag Bishnoi thesis 4 / 28

6 The Fano plane E F D G A B C 7 points and 7 lines, order (2, 2) Anurag Bishnoi thesis 5 / 28

7 The Fano plane 7 points and 7 lines, order (2, 2) Anurag Bishnoi thesis 5 / 28

8 The Fano plane E F D G A B C 7 points and 7 lines, order (2, 2) Anurag Bishnoi thesis 5 / 28

9 The Doily 15 points and 15 lines, order (2, 2) Anurag Bishnoi thesis 6 / 28

10 Automorphism Group Most of the geometries that we study are highly symmetrical. Mathematically, these symmetries are certain rearrangements/permutations of points which map every line to another line. Anurag Bishnoi thesis 7 / 28

11 Automorphism Group Most of the geometries that we study are highly symmetrical. Mathematically, these symmetries are certain rearrangements/permutations of points which map every line to another line. For example, in the square example, look at the permutation A B, B C, C D, D A: D r C C n B l n r m A m B D l A Anurag Bishnoi thesis 7 / 28

12 Automorphism Group Most of the geometries that we study are highly symmetrical. Mathematically, these symmetries are certain rearrangements/permutations of points which map every line to another line. For example, in the square example, look at the permutation A B, B C, C D, D A: D r C C n B l n r m A m B D l A The 4-gon has in total 8 automorphisms (symmetries), the Fano plane has 168 and the Doily has 720. Anurag Bishnoi thesis 7 / 28

13 Near Polygons and Generalized Polygons Near polygons and generalized polygons are two important classes of incidence geometries, which are the central objects of my PhD thesis. Anurag Bishnoi thesis 8 / 28

14 Near Polygons and Generalized Polygons Near polygons and generalized polygons are two important classes of incidence 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 the following axioms: the collinearity graph is connected with finite diameter d; for every point x P and every line L L, there exists a unique point incidence with L which is nearest to x. A generalized n-gon is a point-line geometry whose incidence graph has diameter n and the maximum possible girth, 2n. For n even, every generalized n-gon is a near n-gon. Anurag Bishnoi thesis 8 / 28

15 The Doily A C B l Doily: both a near quadrangle and a generalized quadrangle Anurag Bishnoi thesis 9 / 28

16 The G 2 (4)-near octagon A new incidence geometry of order (2, 10) that has 4,095 points, 15,015 lines and an automorphism group of size 503,193,600. Anurag Bishnoi thesis 10 / 28

17 The G 2 (4)-near octagon A new incidence geometry of order (2, 10) that has 4,095 points, 15,015 lines and an automorphism group of size 503,193,600. For mathematicians: it s a near octagon of order (2, 10) whose automorphism group is isomorphic to a split extension of the finite simple group G 2 (4) by C 2, and it has connections with the Suzuki tower of finite simple groups, L 3 (2) < U 3 (3) < J 2 < G 2 (4) < Suz. Anurag Bishnoi thesis 10 / 28

18 The L 3 (4)-near octagon Another new incidence geometry. It has order (2, 4), 315 points, 525 lines and an automorphism group of size For the mathematicians: it s a near octagon of order (2, 4) whose automorphism group is isomorphic to a split extension of the finite simple group PSL 3 (4) by C 2 C 2 and its points are the nontrivial elations of PG(2, 4) Anurag Bishnoi thesis 11 / 28

19 Historical Context Near polygons introduced in 1980 by Ernie Shult and Arthur Yanushka for studying certain sets of lines in the Euclidean space. Anurag Bishnoi thesis 12 / 28

20 Historical Context Near polygons introduced in 1980 by Ernie Shult and Arthur Yanushka 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 polar spaces, distance regular graphs and sporadic finite simple groups. Anurag Bishnoi thesis 12 / 28

21 Historical Context Near polygons introduced in 1980 by Ernie Shult and Arthur Yanushka 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 polar spaces, distance regular graphs and sporadic finite simple groups. Last new nice near polygon were discovered 15 years ago by Bart De Bruyn as a part of an infinite family. Anurag Bishnoi thesis 12 / 28

22 Semi-Finite Generalized Polygons Tits asked the following question: are there generalized polygons which have finitely many points ( 3) on each line but infinitely many lines through each point? Very little progress has been made over the years, as only the case of generalized quadrangles with 3, 4 or 5 points on each line is solved. Anurag Bishnoi thesis 13 / 28

23 Semi-finite Generalized Hexagons We answer a modified version of the question for the case of generalized hexagons. Theorem Let q {2, 3, 4} and let S be a generalized hexagon isomorphic to the split Cayley hexagon H(q) or its dual H(q) D. Then the following holds for any 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. Anurag Bishnoi thesis 14 / 28

24 Valuation Theory of Near Polygons F I G H I H E G D F E D A B C A B C A 3 3 grid inside a Doily Anurag Bishnoi thesis 15 / 28

25 Some Characterizations Theorem The dual twisted triality hexagon T(2, 8) is the unique near hexagon of order (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 isometrically embedded subgeometry. Theorem The G 2 (4)-near octagon is the unique near octagon of order (2, 10) which contains the Hall-Janko near octagon as an isometrically embedded subgeometry. Anurag Bishnoi thesis 16 / 28

26 POLYNOMIAL METHOD Anurag Bishnoi thesis 17 / 28

27 Covering all but one Given finite sets of real numbers A and B, a finite grid is the set A B R 2 of points with coordinates (x, y) where x A and y B Anurag Bishnoi thesis 18 / 28

28 Covering all but one Given finite sets of real numbers A and B, a finite grid is the set A B R 2 of points with coordinates (x, y) where x A and y B Anurag Bishnoi thesis 18 / 28

29 Covering all but one Given finite sets of real numbers A and B, a finite grid is the set A B R 2 of points with coordinates (x, y) where x A and y B Anurag Bishnoi thesis 18 / 28

30 Covering all but one Given finite sets of real numbers A and B, a finite grid is the set A B R 2 of points with coordinates (x, y) where x A and y B Anurag Bishnoi thesis 18 / 28

31 Covering all but one Given finite sets of real numbers A and B, a finite grid is the set A B R 2 of points with coordinates (x, y) where x A and y B Anurag Bishnoi thesis 18 / 28

32 A polynomial method approach Each line in the plane is given by a linear equation of the form ax + by c = 0, i.e., it is the set of zeros of the polynomial f (x, y) = ax + by c. So, the set of points covered by k lines l 1, l 2,..., l k is equal to the zero set of the degree k polynomial f = f 1 f 2 f k, where f 1,..., f k are the linear polynomials that define the lines. Anurag Bishnoi thesis 19 / 28

33 A polynomial method approach Lemma Given a finite grid A B, if polynomial f (x, y) vanishes on all points of the grid except one then the degree of f is at least A 1 + B 1. Anurag Bishnoi thesis 20 / 28

34 A polynomial method approach Lemma Given a finite grid A B, if polynomial f (x, y) vanishes on all points of the grid except one then the degree of f is at least A 1 + B 1. For mathematicians: of course this also generalizes to n variable polynomials and n-dimensional finite grids A 1 A n. Anurag Bishnoi thesis 20 / 28

35 A more general problem Given k hyperplanes and a finite grid A which is not completely covered by the hyperplanes, how many points do the hyperplanes miss? Anurag Bishnoi thesis 21 / 28

36 A more general problem Given k hyperplanes and a finite grid A which is not completely covered by the hyperplanes, how many points do the hyperplanes miss? Let f (x 1,..., x n ) be an n-variable polynomial of degree d such that there is at least some point (a 1,..., a n ) of A where f (a 1,..., a n ) 0. Find the number of points of A where f does not vanish. Anurag Bishnoi thesis 21 / 28

37 A more general problem Given k hyperplanes and a finite grid A which is not completely covered by the hyperplanes, how many points do the hyperplanes miss? Let f (x 1,..., x n ) be an n-variable polynomial of degree d such that there is at least some point (a 1,..., a n ) of A where f (a 1,..., a n ) 0. Find the number 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 a function of d, A 1,..., A n. Anurag Bishnoi thesis 21 / 28

38 Balls in Bins A 1 A 2 A n Bin A i holds at most A i balls. Anurag Bishnoi thesis 22 / 28

39 Balls in Bins A 1 A 2 A n Bin A i holds at most A i balls. Distribution of k balls is an n-tuple y = (y 1,..., y n ) with y y n = k and 1 y i A i for all i. Anurag Bishnoi thesis 22 / 28

40 Balls in Bins A 1 A 2 A n Let Π(y) = y 1 y n. If n k A A n, let m( A 1,..., A n ; k) be the minimum value of Π(y) as y ranges over all distributions of k balls into bins A 1,..., A n. Anurag Bishnoi thesis 22 / 28

41 Alon-Füredi theorem Theorem (Alon-Füredi) Given a finite grid A = A 1 A n and a polynomial f of degree d which does not vanish on all points of A, there are at least m( A 1,..., A n ; A i d) points of A where f does not vanish. Anurag Bishnoi thesis 23 / 28

42 Alon-Füredi theorem Theorem (Alon-Füredi) Given a finite grid A = A 1 A n and a polynomial f of degree d which does not vanish on all points of A, there are at least m( A 1,..., A n ; A i d) points of A where f does not vanish. Clearly if d < ( A i 1), then we have at least n + 1 balls, and hence the minimum is at least 2. Therefore, the minimum number of hyperplanes required to cover all points of the grid A except one is equal to ( A i 1). Anurag Bishnoi thesis 23 / 28

43 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 an edge contains a 3-regular subgraph Anurag Bishnoi thesis 24 / 28

44 Generalized Alon-Füredi Theorem (Generalized Alon-Füredi Theorem) Let R be a ring and let A 1,..., A n be nonempty finite subsets of R that satisfy Condition (D) a. For i {1,..., n}, let a i = A i and let b i be an integer such that 1 b i a i. Let f R[t 1,..., t n ] be a non-zero polynomial such that deg ti f a i b i for all i {1,..., n}. Let U A = {x A f (x) 0} where A = A 1 A n R n. Then we have U A m(a 1,..., a n ; b 1,... b n ; n a i deg f ). Moreover, for any such R, A 1..., A n and integers b 1,..., b n, we can construct a polynomial f which meets this bound. a For any two distinct elements a, b of the subset, a b is not a zero divisor. i=1 Anurag Bishnoi thesis 25 / 28

45 Chevalley-Warning Theorem Recall that Lemma A polynomial f which vanishes on all points of a finite grid A 1 A n except one, must have degree at least n i=1 ( A i 1). From this lemma, a classical result in mathematics called the Chevalley-Warning theorem directly follows, which is a number theoretic result that has found several applications in combinatorics. Anurag Bishnoi thesis 26 / 28

46 Punctured Chevalley-Warning Theorem Lemma (Ball and Serra) A polynomial f which vanishes on all points of a finite grid A 1 A n except at some point of a subgrid B 1 B n, must have degree at least n i=1 ( A i B i ). Theorem (Punctured Chevalley-Warning) Let f 1,..., f r F q [t 1,..., t n ] be such that r (q 1) deg f j < j=1 n ( A i B i ) i=1 and let Z A be the set of common zeros of the f j s in the grid A. If Z A B, then Z A (A \ B). Anurag Bishnoi thesis 27 / 28

47 Thank you! Anurag Bishnoi thesis 28 / 28