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.
1.A.
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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