Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic Methods for Supersmooth Spline Spaces   Multivariate piecewise polynomial functions (or splines) on polyhedral
complexes have been extensively studied over the past decades and find
applications in diverse areas of applied mathematics including numerical
analysis, approximation theory, and computer aided geometric design. In this
paper we address various challenges arising in the study of splines with
enhanced mixed (super-)smoothness conditions at the vertices and across
interior faces of the partition. Such supersmoothness can be imposed but can
also appear unexpectedly on certain splines depending on the geometry of the
underlying polyhedral partition. Using algebraic tools, a generalization of the
Billera-Schenck-Stillman complex that includes the effect of additional
smoothness constraints leads to a construction which requires the analysis of
ideals generated by products of powers of linear forms in several variables.
Specializing to the case of planar triangulations, a combinatorial lower bound
on the dimension of splines with supersmoothness at the vertices is presented,
and we also show that this lower bound gives the exact dimension in high
degree. The methods are further illustrated with several examples.

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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: eess
9
The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ
3
The categories of this paper are: physics