Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Golod-Shafarevich inequality for Hilbert series of quadratic
  algebras and the Anick conjecture   We study the question on whether the famous Golod-Shafarevich estimate, which
gives a lower bound for the Hilbert series of a (noncommutative) algebra, is
attained. This question was considered by Anick in his 1983 paper 'Generic
algebras and CW-complexes', Princeton Univ. Press., where he proved that the
estimate is attained for the number of quadratic relations $d \leq
\frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the
case for any number of quadratic relations. The particular point where the
number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershik.
He conjectured that a generic algebra with this number of relations is finite
dimensional.
  We prove that over any infinite field, the Anick conjecture holds for
  $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and
confirm the Vershik conjecture over any field of characteristic 0. We give also
a series of related asymptotic results.

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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th