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 Lindstrom 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 Lindstrom valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations. (C) 2017 Elsevier Inc. All rights reserved.