The Number Of Polytopes, Configurations And Real Matroids

Noga Alon

doi:10.1112/S0025579300013875

The Number Of Polytopes, Configurations And Real Matroids

Noga Alon

Research output: Contribution to journal › Article › peer-review

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Abstract

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/d)^d2^n(1+0(1))as n/d→ä. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thom from real algebraic geometry.

Original language	English (US)
Pages (from-to)	62-71
Number of pages	10
Journal	Mathematika
Volume	33
Issue number	1
DOIs	https://doi.org/10.1112/S0025579300013875
State	Published - Jun 1986
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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abstract = "We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/d)d2n(1+0(1))as n/d→{\"a}. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thom from real algebraic geometry.",

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AB - We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/d)d2n(1+0(1))as n/d→ä. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thom from real algebraic geometry.

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The Number Of Polytopes, Configurations And Real Matroids

Abstract

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