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arXiv:1310.6482v5 [math.CO] 29 Mar 2014
EMS Surv. Math. Sci. x (201x), xxx–xxx
DOI 10.4171/EMSS/x
EMS Surveys in
Mathematical Sciences
c
European Mathematical Society
Algebraic combinatorial geometry: the
polynomial method in arithmetic combinatorics,
incidence combinatorics, and number theory
Terence Tao
∗
Arithmetic combinatorics is often concerned with the problem of controlling the p ossible
range of behaviours of arbitrary finite sets in a group or ring with respect t o arithmetic
operations such as addition or multiplication. Similarly, combinatorial geometry is often
concerned with the problem of controlling the possible range of behaviours of arbitrary
finite collections of geometric objects such as points, lines, or circles with respect to
geometric operations such as incidence or distance. Given the presence of arbitrary
finite sets in these problems, the methods used to attack these problems have primarily
been combinatorial in nature. In recent years, however, many outstanding problems in
these directions have been solved by algebraic means (and more specifically, using tools
from algebraic geometry and/or algebraic topology), giving rise to an emerging set of
techniques which is now known as the polynomial method. Broadly speaking, the strategy
is to capture (or at least partition) the arbitrary sets of objects (viewed as points in some
configuration space) in the zero set of a polyn omial whose degree (or other measure of
complexity) is under control; for instance, the degree may be bounded by some function
of the number of objects. One then uses tools from algebraic geometry to understand
the structure of this zero set, and thence to control the original sets of objects.
While various instances of the polynomial method have been known for decades (e.g.
Stepanov’s method, the combinatorial nullstellensatz, or Baker’s theorem), the general
theory of this method is still in the process of maturing; in particular, the limitations
of the polynomial method are not well understood, and there is still considerable scope
to apply deeper results from algebraic geometry or algebraic topology to strengthen the
metho d further. In this survey we present several of the known applications of these
∗
The author is supported by a Simons Investigator grant, the James and Carol Collins Chair,
the Mathematical Analysis & Application Research Fund Endowm ent, and by NSF grant DMS-
1266164. He also thanks Ameera Chowdhury, Ben Green, Felipe Voloch and Michael Zieve
for providing additional examples of the polynomial method, and to Holger Brenner, Fabrice
Orgogozo, Kaloyan Slavov, Jonathan Steinbuch and Andreas Wenz for pointing out errors in
previous versions of the manuscript.
Terence Tao, UCLA Department of Mathematics, Los Angeles, CA 90095-1555.
E-mail: tao@@math.ucla.edu
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