Adjoints of Polytopes: Determinantal Representations and Smoothness

In this article we study determinantal representations of adjoint hypersurfaces of polytopes. We prove that adjoint polynomials of all polygons can be represented as determinants of tridiagonal symmetric matrices of linear forms with the matrix size being equal to the degree of the adjoint. We prove a sufficient combinatorial condition for a surface in the projective three-space to have a determinantal representation and use it to show that adjoints of all three-dimensional polytopes with at most eight facets and a simple facet hyperplane arrangement admit a determinantal representation. This includes all such polytopes with a smooth adjoint. We demonstrate that, starting from four dimensions, adjoint hypersurfaces may not admit linear determinantal representations. Along the way we prove that, starting from three dimensions, adjoint hypersurfaces are typically singular, in contrast to the two-dimensional case. We also consider a special case of interest to physics, the ABHY associahedron. We construct a determinantal representation of its universal adjoint in three dimensions and show that in higher dimensions a similarly structured representation does not exist.

💡 Research Summary

This paper investigates when the adjoint polynomial α_P of a convex polytope P can be written as the determinant of a matrix whose entries are linear forms. The adjoint is defined as the unique homogeneous polynomial of degree k – n – 1 (where k is the number of facets and n the ambient dimension) that vanishes on the residual arrangement R(P) of the facet hyperplanes. This polynomial appears as the numerator of the canonical form in the theory of positive geometries and thus has direct relevance to scattering‑amplitude calculations in physics.

The authors first treat the planar case (n = 2). They prove that for every polygon P, α_P admits a determinantal representation by a symmetric tridiagonal matrix of size equal to the degree of α_P. The matrix is real, positive definite on the interior of the polygon, and its principal minors correspond recursively to adjoints of sub‑polygons. This result is highly restrictive—tridiagonal determinantal representations are rare among plane curves—and it imposes strong topological constraints on the real part of the curve.

Moving to three dimensions, the paper introduces the notion of a “nice arrangement” of lines on a surface of degree D ⊂ ℙ³. The main theorem states that if a surface contains a nice line arrangement of the appropriate degree, then its defining polynomial has a determinantal representation of size D. Using this, the authors show that every three‑dimensional polytope with at most eight facets and a simple facet‑hyperplane arrangement satisfies the condition, and consequently its adjoint polynomial admits a linear determinantal representation. In particular, any such polytope whose adjoint hypersurface is smooth automatically meets the criterion.

For dimensions four and higher, the situation changes dramatically. The authors construct a four‑dimensional polytope whose adjoint hypersurface is smooth, yet they prove that no linear determinantal representation exists. This negative result aligns with classical algebraic‑geometric facts: a smooth hypersurface of degree d ≥ 4 in ℙ³ typically does not contain the special curves required for a determinantal representation (by Noether–Lefschetz theory), and in higher dimensions the obstruction becomes even stronger. Hence, starting from dimension three, only finitely many combinatorial types of polytopes can have smooth adjoints, and from dimension four onward most adjoints lack any linear determinantal representation.

The paper concludes with an application to the ABHY associahedron, a polytope that encodes tree‑level scattering amplitudes in φ³ theory. The authors construct explicit determinantal representations for the universal adjoint in two and three dimensions, but prove that a similarly structured representation cannot exist in dimensions four and above. This underscores a limitation in simplifying higher‑dimensional amplitudes via determinantal formulas.

Overall, the work provides a comprehensive classification of when adjoint polynomials admit linear determinantal representations: a complete positive answer in the plane, a combinatorial‑geometric criterion yielding a positive answer for a broad class of three‑dimensional polytopes, and a series of negative results showing that such representations are generally unavailable in higher dimensions. The results bridge algebraic geometry, combinatorial polytope theory, and the physics of positive geometries, offering both concrete constructions and fundamental obstructions.