Polynomial codimension growth and the Specht problem

We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α ∈(0, 1) a commutative nonassociative algebra A_α whose codimension sequence c_n(A_α), n =1, 2, ..., is polynomially bounded .As an application we are able to construct a new example of a variety with an infinite basis of identities.