Polynomial relations among principal minors of a 4x4-matrix

TL;DR

This paper characterizes the algebraic relations among principal minors of 4x4 matrices, providing explicit polynomial equations that define the variety of principal minors and connecting it to hyperdeterminants.

Contribution

It constructs explicit degree-12 polynomials that define the principal minor variety for 4x4 matrices, completing classical results from the 19th century.

Findings

The principal minor map image is closed.

Explicit degree-12 polynomials define the principal minor variety.

The variety relates to the singular locus of a hyperdeterminant.

Abstract

The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in $\PP^{15}$. The latter is the main component in the singular locus of the 2 x 2 x 2 x 2-hyperdeterminant.