Skip to content
DraconiticResearch
Search Sources API
197,688 studies31 sourcesUpdated just now
PreprintCC BY 4.0arXiv

Clifford Realizations of Hyperbolic Cubics

  • Tim Netzer
Open at sourcePDFDOICite

Abstract

We prove the generalized Lax conjecture for cubics in five variables: their hyperbolicity cones are spectrahedral. Our approach constructs spectrahedral realizations from completely positive maps on Clifford algebras. The construction passes through quaternionic determinantal representations of certain extremal cubics. Convexity of the set of Clifford-realizable cubics then yields the result for all cubics in five variables. The approach also covers all previously known cubic cases of the generalized Lax conjecture: forms in at most four variables and symmetric forms in arbitrarily many variables. We further establish a sufficient criterion for Clifford realizability in arbitrary dimension and use it to obtain a neighborhood of the origin in the cubic norm ball consisting of Clifford-realizable cubics. Finally, we prove spectrahedrality for sparse hyperbolic quartics in four variables.

Related

  1. Canonical analytic realizations of hyperbolic determinantal processes
    Xiang Fang, Feng Guo, Shengzhao Hou +1
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  2. Spin-Clifford Bundles and Spin Connections on the Hyperbolic Clifford Algebra - Part I
    E. Notte-Cuello, J. M. Hoff da Silva
    arXivPreprintarXiv licenceFull textMathematical physicsHigh-energy & particle physicsarXiv
  3. On the Geometric Structure of Hyperbolic Clifford Bundles and Associated Spin Groups
    Eduardo Notte-Cuello
    arXivPreprintarXiv licenceFull textMathematical physicsarXiv
  4. Notes on Obstructions in the Hyperbolic Clifford Algebra Bundle Structure
    Julio Marny Hoff da Silva, Eduardo Notte-Cuello
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  5. Notes on obstructions in the hyperbolic Clifford algebra bundle structure
    J. M. Hoff da Silva, E. Notte-Cuello
    arXivPreprintarXiv licenceFull textMathematical physicsHigh-energy & particle physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{netzer2026clifford,
  title = {{Clifford Realizations of Hyperbolic Cubics}},
  author = {Tim Netzer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12957},
  eprint = {2609.12957},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12957},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Clifford Realizations of Hyperbolic Cubics
AU  - Tim Netzer
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12957
UR  - https://arxiv.org/abs/2609.12957
AB  - We prove the generalized Lax conjecture for cubics in five variables: their hyperbolicity cones are spectrahedral. Our approach constructs spectrahedral realizations from completely positive maps on Clifford algebras. The construction passes through quaternionic determinantal representations of certain extremal cubics. Convexity of the set of Clifford-realizable cubics then yields the result for all cubics in five variables. The approach also covers all previously known cubic cases of the generalized Lax conjecture: forms in at most four variables and symmetric forms in arbitrarily many variables. We further establish a sufficient criterion for Clifford realizability in arbitrary dimension and use it to obtain a neighborhood of the origin in the cubic norm ball consisting of Clifford-realizable cubics. Finally, we prove spectrahedrality for sparse hyperbolic quartics in four variables.
ER  -