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Abstract:
A polynomial in one variable over a noncommutative
algebra may have a lot of different factorizations into a product of
linear factors. The structure of such factorizations can be
rather complicated and the associated Êsubalgebra of pseudo-roots
of the polynomial seems to be very interesting.
One way to describe the situation is to study the directed graph of
the right divisors of the polynomial.
In this talk we present a "universal" approach to this problem.
We introduce and study a wide class of algebras associated
to directed graphs and universal polynomials over these algebras.
We show that for many graphs such algebras are Koszul and compute
their Hilbert series.
The talk is based on joint papers with I. Gelfand, S. Serconek and R.
Wilson.
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