Hierarchical zonotopal spaces

Authors:
Olga Holtz, Amos Ron and Zhiqiang Xu

Journal:
Trans. Amer. Math. Soc. **364** (2012), 745-766

MSC (2010):
Primary 13F20, 13A02, 16W50, 16W60, 47F05, 47L20, 05B20, 05B35, 05B45, 05C50, 52B05, 52B12, 52B20, 52C07, 52C35, 41A15, 41A63

Published electronically:
September 8, 2011

MathSciNet review:
2846351

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Abstract: Zonotopal algebra interweaves algebraic, geometric and combinatorial properties of a given linear map . Of basic significance in this theory is the fact that the algebraic structures are derived from the geometry (via a nonlinear procedure known as ``the least map''), and that the statistics of the algebraic structures (e.g., the Hilbert series of various polynomial ideals) are combinatorial, i.e., computable using a simple discrete algorithm known as ``the valuation function''. On the other hand, the theory is somewhat rigid since it deals, for the given , with exactly two pairs, each of which consists of a nested sequence of three ideals: an external ideal (the smallest), a central ideal (the middle), and an internal ideal (the largest).

In this paper we show that the fundamental principles of zonotopal algebra as described in the previous paragraph extend far beyond the setup of external, central and internal ideals by building a whole hierarchy of new combinatorially defined zonotopal spaces.

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Zonotopal combinatorics,

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Additional Information

**Olga Holtz**

Affiliation:
Department of Mathematics, University of California-Berkeley, Berkeley, California 94720 – and – Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

Email:
holtz@math.berkeley.edu

**Amos Ron**

Affiliation:
Department of Mathematics and Computer Sciences, University of Wisconsin-Madison, Madison, Wisconsin 53706

Email:
amos@cs.wisc.edu

**Zhiqiang Xu**

Affiliation:
LSEC, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

Email:
xuzq@lsec.cc.ac.cn

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05329-8

Keywords:
Zonotopal algebra,
multivariate polynomials,
polynomial ideals,
duality,
grading,
Hilbert series,
kernels of differential operators,
polynomial interpolation,
box splines,
zonotopes,
hyperplane arrangements,
matroids

Received by editor(s):
October 28, 2009

Received by editor(s) in revised form:
February 11, 2010

Published electronically:
September 8, 2011

Additional Notes:
The work of the first author was supported by the Sofja Kovalevskaja Research Prize of Alexander von Humboldt Foundation and by the National Science Foundation under agreement DMS-0635607 and was performed in part at the Institute for Advanced Study, Princeton

The work of the second author was supported by the National Science Foundation under grants DMS-0602837 and DMS-0914986, and by the National Institute of General Medical Sciences under Grant NIH-1-R01-GM072000-01

The work of the third author was supported in part by NSFC grant 10871196 and was performed in part at Technische Universität Berlin

Article copyright:
© Copyright 2011
American Mathematical Society