Implicitisation and Parameterisation in Polynomial Functors

TL;DR

This paper develops algorithms to explicitly compute equations for the closure of images and to parameterize closed subsets within polynomial functors, advancing the understanding of their algebraic structure.

Contribution

It introduces the first algorithms for implicitisation and parameterisation of polynomial functor subsets, making previous theoretical results constructive and computational.

Findings

Algorithm $ extbf{implicitise}$ computes defining equations from a morphism.

Algorithm $ extbf{parameterise}$ constructs morphisms from defining equations.

Both algorithms are effective and implementable.

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.