On GK Dimension and Generator Bounds for a Class of Graded Algebras

In this paper, we introduce the concept of $\mathit{\text{monotonic}}\mathit{\text{algebras}}$, a broad class of algebras that includes all Artin-Schelter regular algebras of dimension at most four, as well as algebras with $\mathit{\text{pure}}$ resolutions, such as Koszul and piecewise Koszul algebras. We show that the Gelfand-Kirillov (GK) dimension of these algebras is bounded above by their global dimension and establish a similar result for the minimal number of generators. Furthermore, we prove a parity theorem for Artin-Schelter regular algebras, demonstrating that the difference between their global dimension and GK dimension is always an even integer.