Sums of squares in convex algebraic geometry

• Applicant: Professor Dr. Claus Scheiderer
• Subject Area: Mathematics
• Term: from 2014 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 253289397

• Final Report Year: 2021

Final Report Abstract

This project was the successor to a previous project, the main result of which had been the disproof of the Helton-Nie conjecture. Some of the main results of the present project concern upper bounds on the complexity of semidefinite representations. In particular, it was proved that every closed convex set in the plane is second-order cone representable. Essentially our proof is even constructive. In joint work with Averkov we tried to extend this result to convex hulls of curves in n-space, the expected result being that they have lifted LMI representations that are block diagonal sums of representations by matrices of size ≤ n+1 . At least for monomial curves this has been shown, and again the approach is constructive. We recon- sidered an important theorem by Helton-Nie, according to which compact convex sets with nonsingular boundary of strict positive curvature are spec- trahedral shadows. We strengthened this result considerably by showing that such sets are even second-order cone representable. On another line, we studied Gram spectrahedra, they parametrize the sums-of-squares repre- sentations of polynomials. Together with two PhD students, we studied the ranks and dimensions of faces and obtained a series of results for quadratic forms on varieties of minimal or almost minimal degree. Generally they concern restrictions on possible dimensions and/or ranks of faces, or other properties of faces like being polyhedral. Finally, in joint work with Kobert we strengthened results from the previous project on polar orbitopes. As one of the consequences, we discovered large classes of previously unknown doubly spectrahedral sets (in fact, all polar orbitopes are of this sort).

Publications

• Extreme points of Gram spectrahedra of binary forms
  C. Scheiderer
  
• Semidefinite representation for convex hulls of real algebraic curves. SIAM J. Appl Algebra Geometry 2, 1–25 (2018)
  C. Scheiderer
  (See online at https://doi.org/10.1137/17M1115113)
• Spectrahedral and semidefinite representability of orbitopes. PhD Thesis, Universität Konstanz, 2018
  T. Kobert
  
• Spectrahedral shadows. SIAM J. Appl Algebra Geometry 2, 26–44 (2018)
  C. Scheiderer
  (See online at https://doi.org/10.1137/17M1118981)
• Spectrahedral representation of polar orbitopes. Manuscripta mathematica
  T. Kobert, C. Scheiderer
  
• Dimensions of Faces of Gram Spectrahedra. PhD Thesis,, Universität Konstanz, 2021
  J. Vill
  
• Polyhedral faces in Gram spectrahedra of binary forms. Linear Algebra and its Applications 608, 133–157 (2021)
  Th. Mayer
  (See online at https://doi.org/10.1016/j.laa.2020.08.025)
• Second-order cone representation for convex sets in the plane. SIAM J. Appl Algebra Geometry 5, 114–139 (2021)
  C. Scheiderer
  (See online at https://doi.org/10.1137/20M133717X)