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We define a GL -variety to be an (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank and played a key role in two recent proofs of Stillman's conjecture. We initiate a systematic study of |$\textbf {GL}$| -varieties and establish a number of foundational results about them. For example, we prove a version of Chevalley's theorem on constructible sets in this setting. [ABSTRACT FROM AUTHOR]

We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindström valuations and Frobenius flocks of such matroids. Building on work by Evans and Hrushovski, we show that the class of algebraic matroids, paired with their Lindström valuations, is not closed under duality of valuated matroids. [ABSTRACT FROM AUTHOR]

In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of co-rank r−1. In this paper, we prove that conjecture for rectangular matrices and for symmetric matrices, as well as a variant for skew-symmetric matrices. [ABSTRACT FROM AUTHOR]