On Projective Equivalence of Univariate Polynomial Subspaces

TL;DR

This paper addresses the problem of classifying univariate polynomial subspaces under projective transformations, using the Wronski map to relate it to binary form equivalence, advancing understanding in polynomial subspace symmetry.

Contribution

It introduces a method to determine projective equivalence of polynomial subspaces by leveraging the Wronski map and group actions, connecting it to binary form classification.

Findings

Established the equivariance of the Wronski map.

Reduced the subspace equivalence problem to binary form equivalence.

Provided a framework for classifying polynomial subspaces under projective transformations.

Abstract

We pose and solve the equivalence problem for subspaces of ${\mathcal P}_n$, the $(n+1)$ dimensional vector space of univariate polynomials of degree $\leq n$. The group of interest is ${\rm SL}_2$ acting by projective transformations on the Grassmannian variety ${\mathcal G}_k{\mathcal P}_n$ of $k$-dimensional subspaces. We establish the equivariance of the Wronski map and use this map to reduce the subspace equivalence problem to the equivalence problem for binary forms.