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Some Applications of Polynomials in Combinatorics
Richard M. Wilson
California Institute of Technology
California, USA |
Abstract
We will present a selection of applications of polynomials and
linear algebra in extremal combinatorics. It is spaces of
multivariate polynomials that will interest us most.
One common method of proving an inequality is to exhibit a set of
$m$ polynomials that can be shown to be linearly independent, but
which belong to the span of $n$ simple polynomials; in this way,
we have proved $m\le n$. We use this technique to derive bounds
on for $L$-intersecting families of sets, codes with few
distances, and two-distance sets in Euclidean space.
Material on the restriction of polynomials to `rectangular'
subsets will be presented and used to prove the Chevelley theorem,
the Erd\H os-Ginzburg-Ziv theorem, the Cauchy-Davenport theorem,
and other results.
If time permits, other topics, such as the $p$-rank of the
incidence matrices of projective planes and extensions of the
Chevelley-Warning theorem, may be introduced.
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| Time: | Wed. May 3, 10:00-11:00
Wed. May 3, 13:30-14:30
Thurs. May 4, 14:00-15:00
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| Place: | Lecture Hall, Niavaran Bldg., Niavaran Sqr., Tehran, Iran |
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