Incidence Geometry and connections to Theoretical Computer Science

1. Incidence Geometry and connections to Theoretical Computer Science ShubhangiSaraf Rutgers University Based on joint works with Albert Ai, ZeevDvir, NeerajKayal, AviWigderson

2. Sylvester-Gallai Theorem (1893) Given: finite number of points in Euclidean plane v v v v Suppose that every line through two points passes through a third

3. Sylvester-Gallai Theorem Given: finite number of points in Euclidean plane v v v v Suppose that every line through two points passes through a third

4. Sylvester-Gallai Theorem • Very well studied. Several proofs known • Several extensions and variations • Complexes, other fields, colorful, quantitative, high-dimensional • Several recent connections to complexity theory • Structure of arithmetic circuits (Polynomial Identity Testing): [Kayal-S 09] • Locally Correctable Codes: [BDWY11, ADSW12, DSW12] • 2-query LCCs over the Reals do not exist • Stable LCCs over Reals do not exists for any constant number of queries

5. Plan • Sylvester-Gallai Theorem and variations • Connections to Polynomial Identity Testing • Connections to Locally Correctable Codes • New proof technique for incidence theorems

6. Sylvester-Gallai Theorem Given a finite set of points S in the plane, that span at least 2 dimensions, then there must be a line through exactly 2 points of S.

7. Proof of Sylvester-Gallai (by Kelly): • By contradiction. If possible, for every pair of points, the line through them contains a third. • Consider the point-line pair with the smallest distance. P m Q ℓ dist(Q, m) < dist(P, ℓ) Contradiction!

8. Points in Complex space Kelly’s Theorem: For every pair of points in , the line through them contains a third, then all points contained in a complex plane Hesse Configuration [Elkies, Pretorius, Swanpoel 2006]: First elementary proof [DSW12]: New proof using basic linear algebra

9. The High Dimensional Sylvester-Gallai Theorem [Hansen 65], [Bonnice-Edelstein 67] • Given a finite set of points in spanning at least 2t dimensions, there exists a t dimensional hyperplane which is spanned by and contains exactly t+1 points. Applications to arithmetic circuits [BDWY11, DSW12]: High dimensional SG over C

10. Quantitative SG For every point there are at least points s.t there is a third point on the line vi Applications to Locally Correctable Codes [BDWY11]: dimension [DSW12]: dimension

11. Stable Sylvester-Gallai Theorem [ADSW12] v Suppose that for every two points there is a third that is -collinear with them v v v

12. Stable Sylvester Gallai Theorem [ADSW12] Applications to stable Locally Correctable Codes v Suppose that for every two points there is a third that is -collinear with them v v v

13. Other extensions • Colorful Sylvester-Gallai Theorem • Average Sylvester-Gallai Theorem • Quantitative, colorful, high-dimensional Sylvester-Gallai

14. Plan • Sylvester-Gallai Theorem and variations • Connections to Polynomial Identity Testing • Connections to Locally Correctable Codes • Proof ideas for some of the incidence theorems

15. Structure of arithmetic circuits

16. Polynomials over R that completely factorize = 0 A B C L11 a1 a2 X1 X2 X3 …

17. = 0 A 0 L11 0 a1 a2 X1 X2 X3 …

18. = 0 Whenever a red and a blue linear function are zero, a green linear function is also zero !! 0

19. Connection to incidence geometry

20. Thm (EK): points must be low dimensional! A+ B+ C= 0 colored points Linear forms of A Linear forms of B Linear forms of C Incidence Configuration: Every line containing points of two colors also includes the third

21. Polynomials depend on only constantly many variables • [KS09] Structure theorem for circuits => first poly time blackbox PIT results for circuits. • Colorful and high dimensional extensions of the Sylvester-Gallaithm

22. Connections to locally correctable codes • Thm[BDWY11, DSW12]: query linear locally correctable codes over R do not exist • Quantitative extensions of Sylvester-Gallai theorem • Thm[ADSW12]: stable linear locally correctable codes over R do not exist for any number of queries • Approximate extensions of the Sylvester-Gallai theorem • Why LCCs over R? • Easier to show lower bounds • Strong enough lower bounds imply arithmetic circuit lower bounds

23. Plan • Sylvester-Gallai Theorem and variations • Connections to Polynomial Identity Testing • Connections to Locally Correctable Codes • Proof ideas for some of the incidence theorems

24. Quantitative SG For every point there are at least points s.t there is a third point on the line vi [BDWY11]: dimension [DSW12]: dimension

25. Rest of the talk: Proof sketch forquant. SG (BDWY11) • Incidence configurations to matrices • Design matrices and Rank bounds for design matrices • Sketch of proof of rank bound • Matrix scaling

26. Incidence configuration to matrices • Given • For every collinear triple , so that • Construct matrix V s.t row is • Construct matrix s.t for each collinear triple there is a row with in positions resp.

28. Rank Bounds to Incidence Theorems • Want: Upper bound on rank of V • How?: Lower bound on rank of A • A turns out to be a design matrix

29. Design Matrices n An m x n matrix is a (k,t)-design matrix if: Each column has at least k non-zeros The supports of every two columns intersect in at most t rows m

30. Theorem (BDWY11, DSW12): Rank Bound Thm(BDWY11, DSW12): Let A be an m x n complex (k,t)-design matrix then: Holds for any field of char=0 (or very large positive char) Not true over fields of small characteristic!

31. Proof of Rank Bound Easy case: All entries are either zero or one m n n Off-diagonals At A = n n m Diagonal entries “perturbed-identity matrix”

32. Bounding the rank of perturbed identity matrices M Hermitian matrix,

33. General Case: Matrix scaling Plan: reduce to easy case using matrix-scaling: c1c2 … cn r1 r2 . . . . . . rm Replace Aij with ricjAij ri, cjpositive reals Same rank, support. • Has ‘balanced’ coefficients:

34. Matrix Scaling Theorem, Sinkhorn (1964) Thm: Let A be a real nx n matrix with non-negative entries. Suppose every zero minor of A of size a x b satisfies Then for every there exists a scaling of A with row sums 1 ± and column sums 1±

35. Sketch of proof • Algorithm (Sinkhorn): Iteratively scale rows and columns of M • doubly stochastic • Claim 1: If is “far” from doubly stochastic, • Claim 2: • Proof (claim 1): • Column scaling • Apply Arithmetic mean/geometric mean inequality, • Permanent is nonzero c1c2 … cn r1 r2 . . . rn

36. Open Questions 3 query LCCs? Geometric translation: Let be a set of points of size such that each point in is contained disjoint coplanar quadruples Then More queries? • [ADSW12]: Bounded coefficient (stable) LCCs over R with any constant number of queries do not exists ``unbounded coefficients?” would imply new arithmetic circuit lower bounds!

37. Thanks!