134 research outputs found
Associated Primes of Spline Complexes
The spline complex whose top homology is
the algebra of mixed splines over the fan
was introduced by Schenck-Stillman in
[Schenck-Stillman 97] as a variant of a complex
of Billera [Billera 88]. In this paper we
analyze the associated primes of homology modules of this complex. In
particular, we show that all such primes are linear. We give two applications
to computations of dimensions. The first is a computation of the third
coefficient of the Hilbert polynomial of , including cases
where vanishing is imposed along arbitrary codimension one faces of the
boundary of , generalizing the computations in [Geramita-Schenck
98,McDonald-Schenck 09]. The second is a description of the fourth coefficient
of the Hilbert polynomial of for simplicial fans
. We use this to derive the result of Alfeld, Schumaker, and Whiteley
on the generic dimension of tetrahedral splines for
[Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description
may be used to give the fourth coefficient in particular nongeneric
configurations.Comment: 40 pages, 10 figure
On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups
Using a result of de Smit and Lenstra, we prove that the order of a group containing nontrivial Gassmann equivalent subgroups must be divisible by at least five primes, not necessarily distinct. We then investigate the existence of Gassmann equivalent subgroups in groups with order divisible by exactly five primes
Regularity of Mixed Spline Spaces
We derive bounds on the regularity of the algebra of
mixed splines over a central polytopal complex
. As a consequence we bound the largest integer
(the postulation number) for which the Hilbert polynomial
disagrees with the Hilbert function
. The polynomial
has been computed in [DiPasquale 2014], building
on [McDonald-Schenck 09] and [Geramita-Schenck 98]. Hence the regularity bounds
obtained indicate when a known polynomial gives the correct dimension of the
spline space . In the simplicial case with all
smoothness parameters equal, we recover a bound originally due to [Hong 91] and
[Ibrahim and Schumaker 91].Comment: 35 pages, 8 figure
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