Implicitisation and Parameterisation in Polynomial Functors

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 GL ∞ -varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 24; no. 5; pp. 1567 - 1593
Main Authors: Blatter, Andreas, Draisma, Jan, Ventura, Emanuele
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2024
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Computer Science
Economics
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Parameterization
Polynomials
14M99
13P99
Polynomial functor
Elimination
Parameterisation
20G05
15A69
Schur functors
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary: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 GL ∞ -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 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 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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-023-09619-6