6533b85efe1ef96bd12bfe1d

RESEARCH PRODUCT

Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras

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We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsEM,L* [1]. We recall that two sequencesan,bnare asymptotically equal, and we writean≃bn,if and only if limn→∞(an/bn)=1.In this paper we prove that\(c_n \left( {M_k \left( G \right)} \right) \simeq c_n \left( {E_{k^2 ,k^2 }^ * } \right) and c_n \left( {M_{k,l} \left( G \right)} \right) \simeq c_n \left( {E_{k^2 + l^2 ,2kl}^ * } \right) \)% MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality\(c_n \left( {M_k \left( F \right)} \right) \simeq c_n \left( {E_{k^2 ,0}^ * } \right) \)% MathType!End!2!1! between the codimensions of the matrix algebraMk(F) and the Capelli polynomials.