Splines, lattice points, and (arithmetic) matroids
Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2014-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2379/pdf |
| _version_ | 1797270314380427264 |
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| author | Matthias Lenz |
| author_facet | Matthias Lenz |
| author_sort | Matthias Lenz |
| collection | DOAJ |
| description | Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by $X$. |
| first_indexed | 2024-04-25T02:02:18Z |
| format | Article |
| id | doaj.art-dca9d46c3bfe4a218e7e472f7fbd3037 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:18Z |
| publishDate | 2014-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-dca9d46c3bfe4a218e7e472f7fbd30372024-03-07T14:53:19ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502014-01-01DMTCS Proceedings vol. AT,...Proceedings10.46298/dmtcs.23792379Splines, lattice points, and (arithmetic) matroidsMatthias Lenz0Mathematical Institute [Oxford]Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by $X$.https://dmtcs.episciences.org/2379/pdflattice polytopebox splinevector partition functiontodd operator(arithmetic) matroidzonotopal algebra[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Matthias Lenz Splines, lattice points, and (arithmetic) matroids Discrete Mathematics & Theoretical Computer Science lattice polytope box spline vector partition function todd operator (arithmetic) matroid zonotopal algebra [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Splines, lattice points, and (arithmetic) matroids |
| title_full | Splines, lattice points, and (arithmetic) matroids |
| title_fullStr | Splines, lattice points, and (arithmetic) matroids |
| title_full_unstemmed | Splines, lattice points, and (arithmetic) matroids |
| title_short | Splines, lattice points, and (arithmetic) matroids |
| title_sort | splines lattice points and arithmetic matroids |
| topic | lattice polytope box spline vector partition function todd operator (arithmetic) matroid zonotopal algebra [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/2379/pdf |
| work_keys_str_mv | AT matthiaslenz splineslatticepointsandarithmeticmatroids |