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

Full article in PDF format | doi: 10.3176/proc.2010.4.08

Natalia Iyudu, Stanislav Shkarin

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  n2/4 and d  n2/2, 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(n2 + 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.


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