Non integral exponential growth of central polynomials

  • Antonio Giambruno
  • Sergey Mishchenko
  • Mikhail Zaicev


Let A be an algebra over a field F of characteristic zero. For every \(n\ge 1\), let \(\delta _n(A)\) be the number of linearly independent multilinear proper central polynomials of A in n fixed variables. It was shown in [8] that if A is a finite dimensional associative algebra, the limit \(\delta (A)=\lim _{n\rightarrow \infty }\root n \of {\delta _n(A)}\) always exists and is an integer. Here we show that such a result cannot be extended in general to non associative algebras. In fact we construct a five-dimensional non associative algebra such that the above limit exists and \(\delta (A)\approx 3.61\), a non integer.


Central polynomial Polynomial identity Codimension Exponential growth 

Mathematics Subject Classification

Primary 16R10 15A69 Secondary 16P90 


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Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PalermoPalermoItaly
  2. 2.Department of Applied MathematicsUlyanovsk State UniversityUlyanovskRussia
  3. 3.Department of Algebra, Faculty of Mathematics and MechanicsMoscow State UniversityMoscowRussia

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