We introduce a many-variable version of the $f$-polynomial and $h$-polynomial associated to a finite simplicial complex. In this context the $h$-polynomial is actually a rational function. We establish connections with the $l^2$-Euler characteristic of right-angled buildings. When $L$ is a triangulation of a sphere we obtain a new formula for the $l^2$-Euler characteristic.
Dan Boros. "$f$-Polynomials, $h$-Polynomials, and $l^2$-Euler Characteristics." Publ. Mat. 54 (1) 73 - 81, 2010.