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 \({\text {GL}}_\infty \)-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 \(\textbf{implicitise}\) that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm \(\textbf{parameterise}\) 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.

在早期的工作中，第二作者表明多项式函子的闭子集总是可以由有限多个多项式方程定义。在\({\text {GL}}_\infty \) -varieties的后续工作中，Bik–Draisma–Eggermont–Snowden 表明，在特征零中，每个这样的闭子集都是态射的图像其域是有限维仿射簇和多项式函子的乘积。在本文中，我们证明这两个结果都可以通过算法实现：存在一种算法\(\textbf{implicitise}\)，它将态射作为多项式函子的输入，并输出有限多个定义图像闭包的方程；以及一种算法\(\textbf{parameterise}\)，该算法将定义多项式函子的闭合子集的有限方程组作为输入，并输出其图像为该闭合子集的态射。