On Circuit Valuation of Matroids

Abstract

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of R ∪ −∞ -valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of R ∪ −∞ -valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

Stiefel tropical linear spaces

“…In fact, any tropical linear space L is equal to the tropical convex hull of its cocircuits [18,29,22], so cocircuits can be thought of as vertices of L from a tropical convexity point of view.…”

Stillman’s conjecture via generic initial ideals

“…Tropical linear spaces are well-studied objects in tropical geometry and matroid theory: the definition above is equivalent to that of [Spe08], except that we allow some coordinates to be ∞. A tropical linear space L gives rise to a matroid M(L) in which the independent sets are those subsets A ⊆ N for which L∩(R A ×{∞} N \A ) = {∞} N , and L is the set of vectors (R-linear combinations of valuated circuits) of a valuated matroid on M(L) [MT01]. With this setup, dim L = |N| − rk(M(L)).…”

Gérard–Levelt membranes

“…For the results listed in this section, we refer to [22], although their definition, following [8], comes with the opposite sign. Let E be a finite set, and a map p) is a valuated matroid if B = ∅ and for B, B ∈ B and u ∈ B\B there exists v ∈ B \B such that…”

“…By orthogonality of circuits and cocircuits ( [22], Theorem 3.11, p. 204), the set of indices that minimise C + C * cannot have cardinality one. Therefore, the contradiction is established.…”