Minimal Hilbert series for quadratic algebras and the Anick conjecture; pp. 301–305

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doi: 10.3176/proc.2010.4.08Authors

Natalia Iyudu, Stanislav Shkarin

Abstract

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* ≤ *n*^{2}/4 and

*d* ≥ *n*^{2}/2, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to

*n*(

*n* – 1)/2 was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.

We announce here the result that over any infinite field, the Anick conjecture holds for *d* ≤ 4(*n*^{2} + *n*)/9 and an arbitrary number of generators. We also discuss the result that confirms the Vershik conjecture over any field of characteristic 0, and a series of related asymptotic results.

References

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