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Zonotopal algebra. (English) Zbl 1223.13010
The present paper’s focus is on combinatorial, geometric, algebraic and analytic properties of low rank linear endomorphisms \(X\) of \(\mathbb R^n\), which are relevant in many areas in mathematics. The paper offers new tools to researchers whose work is at the interplay of algebra, analysis and combinatorics.
Geometric information of \(X\) is captured by its zonotope \(Z(X)\) and its associated hyperplane arrangement \(H(X)\). The authors present in this paper an algebraic theory to study \(X\), \(Z(X)\) and \(H(X)\) based on the association of three algebraic structures (called external, central and internal). Each of these structures consists of a pair of homogeneous polynomials, dual to each other, in the ring of polynomials on \(n\) variables, where \(n\) is the rank of \(X\).
The connections presented in the paper are manifold and demonstrate the power of this algebraic theory and the wide range of possibilities it has to offer. The authors present for example connections between the ideals in the algebraic structures and the zonotope of \(X\) and the integer point in it and its interior. They connect the associated ideals to the number of \(n\)-dimensional regions in the corresponding hyperplane arrangement and also produce good decompositions of the ring of polynomials in \(n\) variables with complex coefficients.
After an introductory section on preliminary results from linear algebra, matroid theory, hyperplane arrangements, zonotopes, polynomial interpolation and polynomial ideals, the main section of the paper shows connections of the introduced algebraic zonotopal theory with group representations, algebraic graph theory, multivariate polynomial interpolation and approximation theory.

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A02 Graded rings
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
41A15 Spline approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
05E40 Combinatorial aspects of commutative algebra
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
47L20 Operator ideals
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