Implicitisation and Parameterisation in Polynomial Functors

Abstract: In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on $\operatorname{GL}\nolimits_{\infty}$-varieties, Bik-Draisma-Eggermont-Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine variety and a polynomial functor. In this paper, we show that both results can be made algorithmic: there exists an algorithm $\mathbf{implicitise}$ that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm $\mathbf{parameterise}$ that takes as input a finite set of equations defining a closed subset of a polynomial functor and outputs a morphism whose image is that closed subset.