The Adjoint Polynomial of a Polyhedral Cone is a Covolume Polynomial

2025/09/24 by Li, Guanxi
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.20606

Abstract

In this paper, we showed that the adjoint polynomial of a polyhedral cone equals the multidegree polynomial of the toric ideal with multigrading, both given by the vertex rays. This fact implies a conjecture of Aluffi, to the effect that the adjoint polynomial of a polyhedral cone in the positive orthant is a covolume polynomial.

Citations

• Linear operators preserving volume polynomials
• Log-concavity of polynomials arising from equivariant cohomology
• Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones
• When are multidegrees positive?
• Projective geometry of Wachspress coordinates

Related

• Canonical forms of polytopes from adjoints
• Geometry of Adjoint Hypersurfaces for Polytopes
• On the derivative cones of polyhedral cones
• Projective geometry of Wachspress coordinates
• Convexity properties of the cone of nonnegative polynomials