Stefan Ovidiu Tohaneanu

arXiv2006

Topological Criteria for $k-$Formal Arrangements

We prove a criterion for $k-$formality of arrangements, using a complex constructed from vector spaces introduced in \cite{bt}. As an application, we give a simple description of $k-$formality of graphic arrangements: Let $G$ be a connected graph with no loops or multiple edges. Let $Δ$ be the flag (clique) complex of $G$ and let $H_{\bullet}(Δ)$ be the homology of the chain complex of $Δ$. If $\mathcal A_G$ is the graphic arrangement associated to $G$, we will show that $\mathcal A_G$ is $k-$formal if and only if $H_i(Δ)=0$ for every $i=1,...,k-1$.

math.CO↗

arXiv2006

Smooth planar $r$-splines of degree $2r$

In \cite{as}, Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $Δ$ (for $d \ge 3r+1$) and any triangulation (for $d\geq 3r+2$). In \cite{ss}, it was conjectured that the Alfeld-Schumaker formula actually holds for all $d \ge 2r+1$. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex $Δ$ such that for any $r$, the dimension of the spline space in degree $d=2r$ is not given by the formula of \cite{as}. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in \cite{ss2}.

math.NA↗