On Projective Equivalence of Univariate Polynomial Subspaces

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 $\mathrm{SL}_2$ 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.