Polynomial Methods in Combinatorial Geometry
نویسندگان
چکیده
iii 1 The Erdős Distance Problem 1 2 Incidence Geometry 5 2.1 The distinct distances incidence problem . . . . . . . . . . . . . . . . . 5 2.2 The geometry of Elekes’ incidence problem . . . . . . . . . . . . . . . . 11 2.3 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Dvir’s Polynomial Method 17 3.1 The Polynomial Method . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Properties of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Proof of the Finite Field Kakeya Conjecture . . . . . . . . . . . . . . . 21 3.4 Extensions to the Method . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 The Joints Conjecture 24 4.1 Algebraic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 The Joints Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Polynomial Partitioning 34 5.1 The Szemerédi-Trotter Theorem . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Decompositions of Space . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 Proof of the Szemerédi-Trotter Theorem via Polynomial Partitioning . 38 6 The Guth-Katz Proof 42 6.1 Proof of the k ≥ 3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 Proof of the k = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.3 Applications to arithmetic combinatorics . . . . . . . . . . . . . . . . . 51 7 The Dirac-Motzkin Conjecture 53 8 Extremal Examples 56 8.1 Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.2 The Böröczky examples . . . . . . . . . . . . . . . . . . . . . . . . . . 59
منابع مشابه
Degenerating Geometry to Combinatorics : Research Proposal for
A central appeal of algebraic geometry is that its functions are polynomials which can be written down as finite expressions. This makes the theory naturally combinatorial, in that polynomials are made up of a finite number of terms which interact through discrete rules. In practice, however, the operations performed on these polynomials are usually too complex for combinatorial methods to be o...
متن کاملFrom combinatorial optimization to real algebraic geometry and back
In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which r...
متن کاملA combinatorial algorithm for computing a maximum independent set in a t-perfect graph
We present a combinatorial polynomial time algorithm to compute a maximum stable set of a t-perfect graph. The algorithm rests on an ε-approximation algorithm for general set covering and packing problems and is combinatorial in the sense that it does not use an explicit linear programming algorithm or methods from linear algebra or convex geometry. Instead our algorithm is based on basic arith...
متن کاملRings with a setwise polynomial-like condition
Let $R$ be an infinite ring. Here we prove that if $0_R$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin X}$ for every infinite subset $X$ of $R$, then $R$ satisfies the polynomial identity $x^n=0$. Also we prove that if $0_R$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in X}$ for every infinite subset $X$ of $R$, then $x^n=x$ for all $xin R$.
متن کاملHodge theory in combinatorics
If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a poly...
متن کاملConvex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.
متن کامل