  Polynomial splines are ubiquitous in the fields of computer aided geometric
design and computational analysis. Splines on T-meshes, especially, have the
potential to be incredibly versatile since local mesh adaptivity enables
efficient modeling and approximation of local features. Meaningful use of such
splines for modeling and approximation requires the construction of a suitable
spanning set of linearly independent splines, and a theoretical understanding
of the spline space dimension can be a useful tool when assessing possible
approaches for building such splines. Here, we provide such a tool. Focusing on
T-meshes, we study the dimension of the space of bivariate polynomial splines,
and we discuss the general setting where local mesh adaptivity is combined with
local polynomial degree adaptivity. The latter allows for the flexibility of
choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on
different faces of the T-mesh. In particular, approaching the problem using
tools from homo-logical algebra, we generalize the framework and the discourse
presented by Mourrain (2014) for uniform bi-degree splines. We derive
combinatorial lower and upper bounds on the spline space dimension and
subsequently outline sufficient conditions for the bounds to coincide.
