961 research outputs found
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
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
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