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Matyas DOMOKOS (Renyi Institute, Budapest). 
Резюме. Kaplansky proved in 1946 that finitely generated associative nil algebras (over an infinite base field) of bounded nil index are nilpotent. For positive integers n and m (and a fixed infinite base field) denote by d(n,m) the minimal positive integer d such that any nil algebra R generated by m elements and having bounded nil index n is nilpotent of nilpotency degree d.
Recent results of Derksen and Makam on generators of rings of matrix invariants together with a theorem of Zubkov from 1996 yield an explicit upper bound on d(n,m) that is polynomial both in n and m and holds for an arbitrary infinite base field (regardless of the characteristic). A lower bound for d(n,2) due to Kuzmin is also extended to positive characteristic.
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