Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The polytope of all $q$-rank functions

Gianira N. Alfarano, Sebastian Degen

TL;DR

This work studies the polytope of all $q$-rank functions for $q$-polymatroids, proving that interior lattice points are absent and that $q$-matroids correspond to lattice vertices. It develops convex combinations of $q$-polymatroids, detailing the impact on flats, cyclic spaces, and $\mu$-independence, with focused results for paving and uniform families. It introduces the characteristic Puiseux polynomial as a broader invariant generalizing the classical characteristic polynomial to rational $q$-polymatroids and derives explicit formulas for certain paving-construction combinations. The paper also explores representable $q$-polymatroids arising from rank-metric codes, showing that representability interacts subtly with convex combinations and is not determined by simple axioms. Overall, it opens several directions—identification of all vertices, rational-vertex theory, and extensions to broader lattice-structural polymatroids—bridging combinatorics, polyhedral geometry, and coding-theory-inspired constructions.

Abstract

A $q$-rank function is a real-valued function defined on the subspace lattice that is non-negative, upper bounded by the dimension function, non-drecreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a $q$-polymatroid. In this paper, we identify these functions with points in a polytope. We show that this polytope contains no interior lattice points, implying that the points corresponding to $q$-matroids are among its vertices. We investigate several properties of convex combinations of two lattice points in this polytope, particularly in terms of independence, flats, and cyclic flats. Special attention is given to the convex combinations of paving and uniform $q$-matroids.

The polytope of all $q$-rank functions

TL;DR

This work studies the polytope of all -rank functions for -polymatroids, proving that interior lattice points are absent and that -matroids correspond to lattice vertices. It develops convex combinations of -polymatroids, detailing the impact on flats, cyclic spaces, and -independence, with focused results for paving and uniform families. It introduces the characteristic Puiseux polynomial as a broader invariant generalizing the classical characteristic polynomial to rational -polymatroids and derives explicit formulas for certain paving-construction combinations. The paper also explores representable -polymatroids arising from rank-metric codes, showing that representability interacts subtly with convex combinations and is not determined by simple axioms. Overall, it opens several directions—identification of all vertices, rational-vertex theory, and extensions to broader lattice-structural polymatroids—bridging combinatorics, polyhedral geometry, and coding-theory-inspired constructions.

Abstract

A -rank function is a real-valued function defined on the subspace lattice that is non-negative, upper bounded by the dimension function, non-drecreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a -polymatroid. In this paper, we identify these functions with points in a polytope. We show that this polytope contains no interior lattice points, implying that the points corresponding to -matroids are among its vertices. We investigate several properties of convex combinations of two lattice points in this polytope, particularly in terms of independence, flats, and cyclic flats. Special attention is given to the convex combinations of paving and uniform -matroids.
Paper Structure (15 sections, 30 theorems, 84 equations)

This paper contains 15 sections, 30 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The polytope of all q-rank functions
  4. The convex combination of two q-(poly)matroids
  5. The flats
  6. The cyclic spaces
  7. The $\mu$-independent spaces
  8. The convex combination of paving q-matroids
  9. The convex combination of uniform q-matroids
  10. Convex combination of two uniform q-matroids
  11. Convex combination of n-2 uniform q-matroids
  12. Cyclic flats
  13. The characteristic Puiseux polynomial
  14. Representable q-polymatroid
  15. Remarks and open questions

Key Result

Proposition 3.5

$\bar{\mathcal{P}}_q^n$ has dimension $T-1$.

Theorems & Definitions (89)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Remark 3.4
  • ...and 79 more