3,172 research outputs found
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Computing Algebraic Matroids
An affine variety induces the structure of an algebraic matroid on the set of
coordinates of the ambient space. The matroid has two natural decorations: a
circuit polynomial attached to each circuit, and the degree of the projection
map to each base, called the base degree. Decorated algebraic matroids can be
computed via symbolic computation using Groebner bases, or through linear
algebra in the space of differentials (with decorations calculated using
numerical algebraic geometry). Both algorithms are developed here. Failure of
the second algorithm occurs on a subvariety called the non-matroidal or NM-
locus. Decorated algebraic matroids have widespread relevance anywhere that
coordinates have combinatorial significance. Examples are computed from applied
algebra, in algebraic statistics and chemical reaction network theory, as well
as more theoretical examples from algebraic geometry and matroid theory.Comment: 15 pages; added link to references, note on page 1, and small
formatting fixe
Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
Using a quantum field theory renormalization group-like differential
equation, we give a new proof of the recipe theorem for the Tutte polynomial
for matroids. The solution of such an equation is in fact given by some
appropriate characters of the Hopf algebra of isomorphic classes of matroids,
characters which are then related to the Tutte polynomial for matroids. This
Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte
polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb.
Series} {\bf B 76} (1999).Comment: 14 pages, 3 figure
Secret-Sharing Matroids need not be Algebraic
We combine some known results and techniques with new ones to show that there
exists a non-algebraic, multi-linear matroid. This answers an open question by
Matus (Discrete Mathematics 1999), and an open question by Pendavingh and van
Zwam (Advances in Applied Mathematics 2013). The proof is constructive and the
matroid is explicitly given
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