The $4 \times 4$ orthostochastic variety
Abstract
Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the real variety were previously known only for $3 \times 3$ matrices. We study the real variety of $4 \times 4$ orthostochastic matrices, and find a minimal defining set of equations consisting of 6 quintics and 3 octics. The techniques used here involve a wide range of both symbolic and computational methods, in computer algebra and numerical algebraic geometry.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 DOI:
 10.48550/arXiv.2001.10691
 arXiv:
 arXiv:2001.10691
 Bibcode:
 2020arXiv200110691C
 Keywords:

 Mathematics  Algebraic Geometry;
 14Q15;
 14P05;
 15B51;
 68W30
 EPrint:
 10 pages (with appendix), 1 ancillary file