Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A lower bound for splines on tetrahedral vertex stars   A tetrahedral complex all of whose tetrahedra meet at a common vertex is
called a \textit{vertex star}. Vertex stars are a natural generalization of
planar triangulations, and understanding splines on vertex stars is a crucial
step to analyzing trivariate splines. It is particularly difficult to compute
the dimension of splines on vertex stars in which the vertex is completely
surrounded by tetrahedra -- we call these \textit{closed} vertex stars. A
formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$
splines on closed vertex stars of degree at least $3r+2$. We show that this
formula is a lower bound on the dimension of $C^r$ splines of degree at least
$(3r+2)/2$. Our proof uses apolarity and the so-called \textit{Waldschmidt
constant} of the set of points dual to the interior faces of the vertex star.
We also use an argument of Whiteley to show that the only splines of degree at
most $(3r+1)/2$ on a generic closed vertex star are global polynomials.

34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
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The categories of this paper are: eess