Algebraic matroids and Frobenius flocks

We show that each algebraic representation of a matroid $M$ in positive characteristic determines a matroid valuation of $M$, which we have named the {\em Lindstr\"om valuation}. If this valuation is trivial, then a linear representation of $M$ in characteristic $p$ can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic $p$ if and only if they are linear in characteristic $p$. 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

A splitter theorem for elastic elements in 33-connected matroids

An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\geq 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and no member of a specific family of $3$-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified $3$-connected minor. We also prove that if $M$ has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.Comment: 22 pages, 2 figure