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Some Contributions to Incidence Geometry and the Polynomial Method

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Some Contributions to Incidence Geometry and the Polynomial Method Some contributions to incidence geometry and the polynomial method Anurag Bishnoi Some contributions to incidence geometry and the polynomial method Anurag Bishnoi Promotor: Prof. Dr. Bart De Bruyn A thesis presented for the degree of Doctor of Sciences Department of Mathematics Ghent University Belgium 2016-2017 Preface The work presented in this thesis falls under two main topics, Incidence Geometry and Polynomial Method. The former deals with incidence structures that are motivated by geometrical notions like projective spaces, affine spaces and polar spaces, while the lat- ter is a loosely defined mathematical technique of using polynomials to solve problems in combinatorics, finite geometry, discrete geometry and number theory. The thesis is accordingly divided into two parts which can be read independently of each other: I Incidence Geometry: Chapters 1 to 6. II Polynomial Method: Chapters 7 to 9. The incidence structures that we will be dealing with in the first part of the thesis are generalized polygons and near polygons, while the polynomial method in the second part is related to zeros of an n variable polynomial over a ring R in certain finite subsets of Rn, which we will call grids. I started my thesis work in August 2013 by learning the theory of valuations of generalized polygons which was developed by my supervisor Bart De Bruyn to obtain characteriza- tions of small generalized polygons. He had successfully applied this theory to prove a spectacular result that the Ree-Tits octagon of order (2; 4) is the unique generalized oc- tagon of order (2; 4) which contains a suboctagon of order (2; 1). The basic idea in this theory is to first compute certain integer valued functions on the point set of a given generalized polygon, the so-called valuations, and a particular incidence structure formed by these valuations called the valuation geometry of the generalized polygon. This valu- ation geometry is then used to obtain the required information about every generalized polygon that contains the given generalized polygon as a full subgeometry. He saw fur- ther potential in this theory and gave me a result on semi-finite generalized hexagons, which he knew how to prove, as a “toy problem” that I could work on after mastering the techniques that he had developed. At around November 2013, I successfully solved this problem, which I consider as my first milestone in mathematical research. We had proved that (a) no (possibly infinite) generalized hexagon can contain the split Cayley hexagon H(2) as a proper full subgeometry, and (b) every generalized hexagon containing the dual split Cayley hexagon H(2)D as a proper full subgeometry is finite, and hence isomorphic to the dual twisted triality hexagon T(2; 8) by the classification result of Co- hen and Tits. Within a couple of months we generalized the techniques further so that we could prove similar results for near hexagons (which are more general than general- ized hexagons) containing these subhexagons. This work became my first research paper. We have recently extended our results further by considering generalized hexagons with q + 1 2 f4; 5g points on each line containing one of the known generalized hexagons of order q as a subgeometry. All of these results constitute Chapter 4 of this thesis. As an offshoot of our work on semi-finite generalized polygons, Bart and I thought it would be nice to compute the valuation geometry of some other well-known near polygons with Page v three points on each line (many of our techniques were limited to point-line geometries with three points on each line). The most natural choice for us was the Hall-Janko near octagon HJ, which corresponds to the sporadic finite simple group J2 and contains subgeometries isomorphic H(2)D. I computed the valuation geometry of HJ, and as a result we noticed that there might exist a near octagon with three points on each line containing HJ which can be constructed using certain valuations of HJ. Since there was no known example of such a near octagon, this was unexpected (and exciting!). I checked, using a computer, that indeed we have a near octagon containing HJ, and quickly started finding its properties with the help of the mathematical software SageMath. My computations revealed that the automorphism group of this new near octagon acted vertex-transitively on the collinearity graph, which was a bit surprising due to the asymmetrical nature of our construction using certain valuations of HJ. Moreover, SageMath gave me the order of the full automorphism group and the fact that its derived subgroup is a simple group. From this we conjectured that the automorphism group of our new near octagon must be isomorphic to G2(4):2, where G2(4) is the well-known finite simple group of Lie type. We finally arrived at a group theoretical construction of this near octagon using central involutions of G2(4):2 and proved our conjecture. Sitting inside this new near octagon, we discovered another new near octagon, which corresponds to the finite simple group L3(4) (the projective special linear group). Both of these near octagons are new and they share many common properties; for example, both of them can be defined by a suborbit diagram of central involutions which look quite similar. We give a common treatment of these two new incidence geometries in Chapter 5, where we derive several important structural properties. The new near octagon corresponding to the group G2(4), let’s call it O1, has HJ as a subgeometry, which in turn has H(2)D has a subgeometry. In this manner we get a D “tower” of near polygons, H(2; 1) ⊂ H(2) ⊂ HJ ⊂ O1, where H(2; 1) is the unique generalized hexagon of order (2; 1) (it is the dual of the incidence graph of Fano plane). The automorphism groups of these near polygons are L3(2):2, U3(3):2, J2:2 and G2(4):2, respectively. Thus we have the first four members of the Suzuki tower of finite simple groups, L3(2) < U3(3) < J2 < G2(4) < Suz, where Suz is the sporadic simple group discovered by Michio Suzuki in 1969. Suzuki constructed Suz as an automorphism group of the largest graph in a sequence Γ0;:::; Γ4 of graphs in which Γ1;:::; Γ4 are strongly regular graphs and for each i 2 f0;:::; 3g, the graph Γi is the local graph of Γi+1. Using our Suzuki tower of near polygons, we can show that all of these graphs can be constructed in a uniform geometrical fashion. Moreover, we have proved that every near polygon in this tower, except the first one, is the unique near polygon of its order and diameter containing the previous near octagon as a subgeometry. These results constitute Chapter 6 of the thesis. A significant part of my work in incidence geometry involved computations on computer models of near polygons and generalized polygons using mathematical software like GAP and SageMath. We had to design reasonably fast ways of finding certain substructures of near polygons called hyperplanes, which would then be used to construct the valuation geometry of a given near polygon. While working on these algorithms, I realised that that they can help us answer some open questions on distance-j ovoids of generalized polygons. In collaboration with Ferdinand Ihringer, I was able to show non-existence of distance-2 ovoids in the dual split Cayley hexagon H(4)D. The techniques that we developed for this work are quite general and they might be helpful in resolving other such “small cases” of Page vi open problems in finite geometry. All my computational work, some of which is required in subsequent chapters, is contained in Chapter 3. Towards the end of 2014, I started getting interested in the so-called polynomial method after reading Terence Tao’s blog post on Zeev Dvir’s two page solution of the famous finite field Kakeya problem using a slick polynomial argument. I quickly learned about Noga Alon’s Combinatorial Nullstellensatz, Tom H. Koornwinder’s proof of the absolute bound on equiangular lines, Andries Brouwer and Lex Schrijver’s proof of the bound on affine blocking sets, and several other such important results involving (elementary) polynomial arguments. By March 2015, I started writing some notes on the polynomial method in collaboration with my friends Abhishek Khetan and Aditya Potukuchi who were also interested in understanding these techniques. We read several papers and ex- pository articles to get acquainted with some of the basic ideas involved. In particular, we started reading the papers “Combinatorial Nullstellensatz Revisited” by Pete Clark and “Warning’s Second Theorem with Restricted Variables” by Pete Clark, Aden Forrow and John Schmitt. In April 2015, I emailed Pete and John saying that one can easily obtain a generalization of the classical Chevalley-Warning theorem (which is a corollary of Warn- ing’s second theorem) using a result of Simeon Ball and Oriol Serra called the Punctured Combinatorial Nullstellensatz. My new result is in fact a generalization of David Brink’s restricted variable Chevalley-Warning theorem, which he proved using Alon’s Combinato- rial Nullstellensatz. We give the exact statement in Chapter 8 where we also lay down the basics of grid reduction of polynomials and give a new proof of Punctured Combinatorial Nullstellensatz. Pete and John responded promptly to my email(s) and shared with me a lot of material related to the things they have been working on. The discussions that followed with Pete, John and Aditya resulted in a collaborative work which I believe is my main contribution to the polynomial method.
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