3 research outputs found
Primary Decomposition with Differential Operators
We introduce differential primary decompositions for ideals in a commutative
ring. Ideal membership is characterized by differential conditions. The minimal
number of conditions needed is the arithmetic multiplicity. Minimal
differential primary decompositions are unique up to change of bases. Our
results generalize the construction of Noetherian operators for primary ideals
in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method
for representing affine schemes. The case of modules is also addressed. We
implemented an algorithm in Macaulay2 that computes the minimal decomposition
for an ideal in a polynomial ring
Zonotopal algebra and forward exchange matroids
Zonotopal algebra is the study of a family of pairs of dual vector spaces of
multivariate polynomials that can be associated with a list of vectors X. It
connects objects from combinatorics, geometry, and approximation theory. The
origin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes the
Dahmen-Micchelli space that is spanned by the local pieces of the box spline
and P(X) is a space spanned by products of linear forms.
The first main result of this paper is the construction of a canonical basis
for D(X). We show that it is dual to the canonical basis for P(X) that is
already known.
The second main result of this paper is the construction of a new family of
zonotopal spaces that is far more general than the ones that were recently
studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We
call the underlying combinatorial structure of those spaces forward exchange
matroid. A forward exchange matroid is an ordered matroid together with a
subset of its set of bases that satisfies a weak version of the basis exchange
axiom.Comment: 34 pages, 4 figures, minor corrections, same as journal version (up
to layout