Coordinate-wise Powers of Algebraic Varieties

TL;DR

This paper introduces coordinate-wise powers of algebraic varieties, analyzing their degrees and equations, and applies these concepts to orthostochastic matrices, dual varieties, and real symmetric matrices with degenerate spectra.

Contribution

It defines and studies coordinate-wise powers of varieties, providing formulas for degrees and equations, and explores applications to matrix varieties and dual varieties.

Findings

Degree formulas for coordinate-wise powers

Explicit equations for hypersurfaces and linear spaces

Connections to orthostochastic matrices and symmetric matrices

Abstract

We introduce and study coordinate-wise powers of subvarieties of $\mathbb{P}^n$, i.e. varieties arising from raising all points in a given subvariety of $\mathbb{P}^n$ to the $r$-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of $\mathbb{P}^n$ under the quotient of $\mathbb{P}^n$ by the action of the finite group $\mathbb{Z}_r^{n+1}$. We determine the degree of coordinate-wise powers and study their defining equations, particularly for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.