USING (POLYNOMIAL) CELL DECOMPOSITIONS
1. Szemerdi-Trotter
e
We will recall the standard form of the theorem.
Theorem 1.1. If S is a set of S points and L is a set of L lines (all in R2 ), then
the number of incidences obeys the following bound:
I (S, L)
REGULI AND APPLICATIONS, ZARANKIEWICZ PROBLEM
One of our long-term goals is to pursue some questions of incidence geometry in
R . We recall one question to direct our focus during this lecture.
3
Question 1. If L is a set of L lines in R3 , and if at most
The Polynomial Method
Professor Larry Guth
Friday, October 5
Today will be the last background lecture in incidence geometry. We nish it o with a discussion
of a cool new way to approach the distinct distance problem; this approach, due to Elekes, has
con
ALGEBRAIC STRUCTURE AND DEGREE REDUCTION
Let S Fn . We dene deg (S ) to be the minimal degree of a non-zero polynomial
that vanishes on S . We have seen that for a nite set S , deg (S ) n|S |1/n . In fact,
we can say something a little sharper. Let V (d)
APPLICATION TO INCIDENCE THEORY OF LINES IN SPACE
In connection with the distinct distance problem, we encountered the following
question about lines in R3 . Given L lines in R3 with L1/2 lines in any plane, how
many points can there be in Pk (L)? Elekes
BEZOUT THEOREM
One of the most fundamental results about the degrees of polynomial surfaces is
the Bezout theorem, which bounds the size of the intersection of polynomial surfaces.
The simplest version is the following:
Theorem 0.1. (Bezout in the plane)
CROSSING NUMBERS AND DISTINCT DISTANCES
The Szemerdi-Trotter theorem plays a fundamental role in incidence geometry in
e
the plane. It can be rephrased in several equivalent ways, and it helps to know the
dierent ways. We recall three standard phrasings h
The distinct distance problem and the unit distance
problem
During this lecture, we examine the distinct distance problem and the unit distance
problem. We would like to apply the following theorem about crossing numbers, proven last
time:
Theorem 1. If G
THE BERLEKAMP-WELCH ALGORITHM
Suppose that we have a polynomial P of fairly low degree over a nite eld F. The
data is corrupted, leaving a function F : F F, and we know that F (x) = P (x)
for a certain fraction of x F. We want to understand whether we can
THE JOINTS PROBLEM
Suppose that we have a set of lines in R3 . A joint of the set of lines is a point
which lies in three non-coplanar lines. The joints problem asks what is the maximal
number of joints that can be formed from L lines.
Lets look at some e
WHY POLYNOMIALS? PART 1
The proofs of nite eld Kakeya and joints and short and clean, but they also
seem strange to me. They seem a little like magic tricks. In particular, what is the
role of polynomials in these problems? We will consider this question
CROSSING NUMBERS AND THE SZEMEREDI-TROTTER
THEOREM
In this lecture we study the crossing numbers of graphs and apply the results
to prove the Szemeredi-Trotter theorem. These ideas follow the paper Crossing
numbers and hard Erds problems in discrete geom
Pesto
1
Fall 2012
1
Incidence Geometry
Topic: take a bunch of simple shapes like circles or lines, and study how they can intersect each
other.
Denition 1.1. If L is a set of |L| lines in R2 , let Pk (L) be the set of points lying in at least k
lines, cal