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Bounds on the number of cells and the dimension of the Dressian
Combinatorics · Submitted 18 Aug 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$.
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The abstract, sentence by sentence
The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We present upper bounds on the number of cells and the dimension of $\mathcal{Dr}(M)$. For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$. For uniform matroids $M=U(r,n)$, these upper bounds are comparable to lower bounds derived from valuations that are constructed from sparse paving matroids.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s1 | This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. | background 1.00 | 0.18 | 0.04 | asserted 1.00 |
| s2 | We present upper bounds on the number of cells and the dimension of $\mathcal{Dr}(M)$. | contribution 0.97 | 0.94 | 0.34 | asserted 1.00 |
| s3 | For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$. | result 1.00 | 0.96 | 0.89 | asserted 1.00 |
| s4 | For uniform matroids $M=U(r,n)$, these upper bounds are comparable to lower bounds derived from valuations that are constructed from sparse paving matroids. | result 0.55 | 0.83 | 0.10 | asserted 0.91 |
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Original abstract
The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We present upper bounds on the number of cells and the dimension of $\mathcal{Dr}(M)$. For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$. For uniform matroids $M=U(r,n)$, these upper bounds are comparable to lower bounds derived from valuations that are constructed from sparse paving matroids.
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