Splines, lattice points, and arithmetic matroids

@article{Lenz2014SplinesLP,
  title={Splines, lattice points, and arithmetic matroids},
  author={Matthias Lenz},
  journal={Journal of Algebraic Combinatorics},
  year={2014},
  volume={43},
  pages={277-324}
}
  • Matthias Lenz
  • Published 2014
  • Mathematics
  • Journal of Algebraic Combinatorics
Let X be a $$(d\times N)$$(d×N)-matrix. We consider the variable polytope $$\varPi _X(u) = \{ w \ge 0 : X w = u \}$$ΠX(u)={w≥0:Xw=u}. It is known that the function $$T_X$$TX that assigns to a parameter $$u \in \mathbb {R}^d$$u∈Rd the volume of the polytope $$\varPi _X(u)$$ΠX(u) is piecewise polynomial. The Brion–Vergne formula implies that the number of lattice points in $$\varPi _X(u)$$ΠX(u) can be obtained by applying a certain differential operator to the function $$T_X$$TX. In this article… Expand
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References

SHOWING 1-10 OF 56 REFERENCES
Interpolation, Box Splines, and Lattice Points in Zonotopes
  • 6
  • PDF
Lattice Points in Polytopes, Box Splines, and Todd Operators
  • 5
  • PDF
Box splines and the equivariant index theorem
  • 23
  • PDF
Vector partition functions and index of transversally elliptic operators
  • 31
  • Highly Influential
  • PDF
Residue formulae for vector partitions and Euler-MacLaurin sums
  • 94
  • PDF
Residue formulae, vector partition functions and lattice points in rational polytopes
  • 151
  • PDF
Zonotopal algebra and forward exchange matroids
  • 12
  • PDF
Zonotopal algebra
  • 42
  • Highly Influential
  • PDF
The many aspects of counting lattice points in polytopes
  • 61
  • PDF
Hierarchical zonotopal spaces
  • 17
  • Highly Influential
  • PDF
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