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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
DOI: https://doi.org/10.1090/S0002-9947-2011-05329-8
Published electronically: September 8, 2011
MathSciNet review: 2846351
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Abstract | References | Similar Articles | Additional Information

Abstract: Zonotopal algebra interweaves algebraic, geometric and combinatorial properties of a given linear map $ X$. 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 $ X$, 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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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: https://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

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