Abstract
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from Question 7 in Ranestad and Sturmfels (Le Math. 75, 411–424, 2020) and Problem 13 in Sturmfels (2014), which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a Julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.
Highlights
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d
Many mathematical models are defined by nonlinear maps f : V → W between vector spaces
The inverse problem is to decide if a point w ∈ W belongs to the image of f, and if so, to determine its preimage
Summary
Many mathematical models are defined by nonlinear maps f : V → W between vector spaces. In this article we focus on the case when f is a polynomial map Under this assumption the forward problem consists in evaluating a system of polynomials, and the inverse problem is to solve a system of polynomial equations. We make the following assumption for f : we require it to be mapping into a vector space of polynomials and we assume that the image of f is GL-invariant. Our motivation comes from Question 7 in [31] and Problem 13 in [33], which ask to find equations for varieties of cubic and quartic symmetroids These are subvarieties of the vector space of homogeneous polynomials in n = 4 variables of degree respectively c = 3 and c = 4. The space of polynomials vanishing is often huge, but the GL action reduces the complexity and allows us to describe it using just a few generators
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