On structure of domains with quadratic growth

Let F be a finite field, and let R be an affine F-algebra which is a domain of Gelfand-Kirillov dimension smaller than 3. Let m,n be natural numbers. Assume that x∈R is transcendental over F and y1,…,yn∈R are such that ∑i,j⩽mαi,jxiykxj=0, for some αi,j∈F (not all equal to 0) and each k⩽n. It is shown that either R satisfies a polynomial identity or else the subalgebra of R generated by y1,y2,…,yn and x has Gelfand-Kirillov dimension 1. From this we deduce that a finitely generated domain over F with quadratic growth and with an infinite centre satisfies a polynomial identity (is a PI domain). Moreover, the centralizer of a non-algebraic element in a finitely generated domain with quadratic growth over finite field is a PI domain.