Graded algebras

We study the growth in the number of dimensions $d_n$ of the homogeneous component of a graded algebra with a finite number of defining relations and generators for the Poincaré series $\sum d_nx^n$. It is proved that if the defining relations are words, the Poincaré series is a rational function. In the general case inequalities are proved linking the number of dimensions $d_n$ with the number of generators defining relations and their degree.