发表机构
University of Innsbruck（因斯布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明五变量三次型的广义Lax猜想，通过Clifford代数上的完全正映射构造谱面体实现，并推广至稀疏四次型。
AI 中文摘要
我们证明了五变量三次型的广义Lax猜想：其双曲锥是谱面体。我们的方法通过Clifford代数上的完全正映射构造谱面体实现。该构造经由某些极值三次型的四元数行列式表示。Clifford可实现三次型集合的凸性进而得出五变量所有三次型的结果。该方法还涵盖了广义Lax猜想所有先前已知的三次型情形：至多四个变量的形式以及任意多个变量的对称形式。我们进一步建立了任意维数下Clifford可实现性的充分判据，并利用它获得三次范数球中由Clifford可实现三次型组成的原点邻域。最后，我们证明了四变量稀疏双曲四次型的谱面体性。
英文摘要
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.
