4,349 research outputs found
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
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
A note on algebraic rank, matroids, and metrized complexes
We show that the algebraic rank of divisors on certain graphs is related to
the realizability problem of matroids. As a consequence, we produce a series of
examples in which the algebraic rank depends on the ground field. We use the
theory of metrized complexes to show that equality between the algebraic and
combinatorial rank is not a sufficient condition for smoothability of divisors,
thus giving a negative answer to a question posed by Caporaso, Melo, and the
author.Comment: To appear in Mathematical Research Letter
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
Characteristic Sets of Matroids
Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set.
The linear representations and characteristic sets are well studied. But the algebraic representations and characteristic sets received much less attention, and the possible algebraic characteristic sets are still not completely known. This dissertation is a study of possible pairs of linear-algebraic characteristic sets of matroids.
Furthermore, if a matroid has an algebraic representation over a positive characteristic field, then the matroid can be represented by a particular set of linear matroids in a field of the same characteristic, called Frobenius flock. In this dissertation, we also have studied Frobenius flock representations, and possible flock characteristic sets
Algebraic matroids and Frobenius flocks
We show that each algebraic representation of a matroid in positive
characteristic determines a matroid valuation of , which we have named the
{\em Lindstr\"om valuation}. If this valuation is trivial, then a linear
representation of in characteristic can be derived from the algebraic
representation. Thus, so-called rigid matroids, which only admit trivial
valuations, are algebraic in positive characteristic if and only if they
are linear in characteristic .
To construct the Lindstr\"om valuation, we introduce new matroid
representations called flocks, and show that each algebraic representation of a
matroid induces flock representations.Comment: 21 pages, 1 figur
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