Vanishing Identities for Rational Polytopes

• The paper presents novel vanishing identities that uniquely characterize adjoint polynomials through higher-order interpolation along residual arrangements.
• It leverages factorization structures to extend classical Vandermonde identities, providing combinatorial criteria for constructing Delzant and rational Delzant polytopes.
• The research connects vanishing phenomena in Ehrhart theory, mixed volume invariants, and discrete moments, offering new computational and classification tools for polytopes.

Novel vanishing identities for rational polytopes are a recent and systematically emerging phenomenon at the interface of algebraic, geometric, and combinatorial aspects of convex polytope theory. Modern research has revealed multiple, highly structured vanishing laws ranging from interpolation in residual schemes to generalised Vandermonde determinants, mixed-volume constraints, and lattice invariants. These identities are central to the understanding of canonical structures (such as adjoint polynomials, Ehrhart polynomials, and algebraic degrees), the behavior of generating functions, and to the precise characterization of certain distinguished classes of polytopes (reflexive, Gorenstein, dual-defect, Delzant, etc.).

1. Adjoint Polynomials and Residual Vanishing

For a full-dimensional convex polytope $P \subset \mathbb{P}^n$ with facets cut out by linear forms $\{l_i\}$, Brüser–Weigert established that the adjoint polynomial $\mathrm{adj}_P$ of $P$ is, up to scaling, uniquely determined as the homogeneous polynomial of degree $d-n-1$ vanishing to a prescribed order on each residual linear subspace $L$ associated with the facet arrangement. The order of vanishing $\mathrm{ord}_P(L)$ is determined combinatorially via the configuration of intersecting facets and their incidence with $P$. Explicitly:

$$\mu_{L}(f) \geq \mathrm{ord}_P(L) \quad \forall\,\text{residual flat } L \subset \mathbb{P}^n \implies f \propto \mathrm{adj}_P.$$

Notably, it suffices to impose the vanishing only at the zero-dimensional residual points ($R_0(P)$), with appropriate multiplicities. This higher-order interpolation formulation resolves a fundamental question posed by Kohn and Ranestad regarding the characterization and uniqueness of the adjoint beyond the simple arrangement case (Brüser et al., 17 Nov 2025).

The geometric and computational consequences are significant: the singularities of the adjoint hypersurface $A_P = \{\mathrm{adj}_P = 0\}$ occur at precisely the predicted orders along residual arrangements, and for practical computation, the system reduces to solving a finite linear system determined by $R_0(P)$ and associated multiplicities.

2. Generalised Vandermonde Identities from Factorization Structures

Factorization structures, essential in toric, Sasakian, and Kähler geometry, produce algebraic families of polytopes with deep combinatorial symmetries. These structures enable the formulation of generalised Vandermonde (vanishing) identities. For a φ-compatible polytope—whose facet normals lie on factorization curves—one has, for any generic selection of m + 1 points on these curves, the basic vanishing condition:

$$\langle \varphi^T(x),\,\psi_j(\ell_j)\rangle = 0, \qquad j = 1,\ldots,m+1,$$

where $x = \ell_1 \otimes \cdots \otimes \ell_m$ and the ψ_j are the factorization curves. Higher-order analogues, obtained by differentiating with respect to the parameters, yield multilinear vanishing identities, generalising the classical Vandermonde determinant to the context of cyclic, Segre–Veronese, and more general product polytopes.

In the case of Veronese (cyclic) polytopes, these identities reduce to systems involving elementary symmetric functions and are at the heart of explicit constructions of Delzant and rational Delzant polytopes. The generalised identities also encode a combinatorial criterion (Gale evenness) determining which sets of facet normals support faces, and provide necessary and sufficient conditions for all normals to lie in a common full-rank lattice (Púček, 2023).

3. Vanishing in Newton and Mixed Volume Invariants

The Newton number $\nu(P)$ of a convenient lattice polytope is the alternating sum of the volumes of coordinate subspace intersections. A central vanishing identity is the classification of all polytopes for which $\nu(P) = 0$, characterized as Bₖ-polytopes—Cayley sums satisfying specific dimension inequalities among summands. For such polytopes, all mixed volume terms that appear in the Kouchnirenko formula vanish:

$$\nu_C(P_0 * \cdots * P_k) = \sum_{\substack{a_0,...,a_k > 0 \ a_0 + \cdots + a_k = n-k}} MV(\underbrace{P_0,...,P_0}_{a_0},..., \underbrace{P_k,...,P_k}_{a_k}) = 0.$$

Additionally, inclusion-exclusion type identities among lattice volumes of Cayley sums provide alternate vanishing formulas:

$$\sum_{I \subset \{1,...,k\}} (-1)^{k-|I|} \operatorname{Vol}_{\mathbb{Z}}(P_I) = 0, \qquad P_I := P_0 * \left(*_{i \in I} P_i\right).$$

These vanishing laws directly control the behavior of algebraic invariants (e.g., algebraic, maximum likelihood, and Euclidean distance degrees via the e-Newton number) and supply a partial solution to Arnold's monotonicity problem (Selyanin, 4 Jul 2025). The thinness phenomenon—meaning polytopes not covered by the classical BKK theory—arises precisely for these vanishing polytopes, with Bₖ-polytopes forming a broad new class of such examples.

4. Boundary $h^*$-Polynomials, Ehrhart Reciprocity, and Vanishing at Roots of Unity

Classical Ehrhart theory extends to rational polytopes with denominator $q$ by interpreting the Ehrhart (quasi-)polynomial as a rational generating function with numerator $h^*_P(z)$. Rational reflexivity and Gorenstein properties enforce rigid vanishing conditions for $h^*_P(z)$ at roots of unity:

• If $P$ is rational reflexive of denominator $q$, then for all $\zeta^q = 1,\, \zeta \neq 1$:

$h^*_P(\zeta) = 0.$

• For rational $g$-Gorenstein $P$:

$$h^*_P(\zeta) = 0 \quad \textrm{whenever } \zeta^q = 1 \text{ or } \zeta^g = 1,\, \zeta\neq 1.$$

• For Ehrhart dilations, vanishing identities force affine-linear relations between the boundary h*-polynomial and interior datum evaluated at appropriate roots.

These phenomena are precisely formulated via palindromic symmetric decompositions of $h^*_P(z)$ into combinations of boundary and interior data, grounded in half-open simplex triangulations, with functional vanishing conditions arising from the roots of unity inherent in reflexivity and dilation structure (Bajo et al., 2022).

5. Barnes Polynomial Identities and Discrete Moments

For simple integer polytopes, novel vanishing identities have been obtained through expansions of the integer-point generating functions using Barnes polynomials and discrete moments of half-open parallelepipeds. The core identity for $P \subset \mathbb{R}^d$ is:

$$0 = \sum_{k=0}^m \binom{m}{k} \sum_{v \in V} B_k\big(t \langle v, z \rangle, \mathbf{a}_v\big) \sum_{p \in \Pi_v \cap \mathbb{Z}^d} \langle p, z \rangle^{m - k}, \quad m=0,...,d-1, \ \forall t>0,$$

where $V$ is the vertex set, $w_k(v)$ are outgoing integral edge-vectors, $$\mathbf{a}_v = ( \langle w_1(v), z \rangle, ..., \langle w_d(v), z \rangle )$$, and $B_k$ denotes the Barnes polynomials. The holomorphy of the generating function in the auxiliary variable enforces the vanishing of negative-power terms, and thus these identities generalize Brion–Lawrence and Brion–Vergne volume and reciprocity formulas. The identities also provide linear constraints among Barnes numbers and discrete moments, reflecting deep relations among the local and global polytope data (Robins, 18 Jan 2026).

6. Degrees of Vanishing and Asymptotic Mixed Volume Constraints

For a bounded convex (possibly rational) polytope $P \subset \mathbb{R}^n$, the minimal vanishing degree $r_P$ and interpolation degree $s_P$ for polynomials on $P \cap \mathbb{Z}^n$ satisfy asymptotic limits as dilations grow large:

$$v_P = \lim_{d\to\infty} \frac{r_{dP} - 1}{d}, \qquad w_P = \lim_{d\to\infty} \frac{s_{dP}}{d},$$

with the mixed-volume bound

$v_P^{n-1} w_P \leq n! \cdot \operatorname{vol}(P).$

In the case $P$ is a triangle, equality holds. These invariants capture the vanishing and interpolation complexity at the lattice level and systematize the combinatorics of standard monomial sets, leading to powerful new combinatorial and algebraic vanishing results (Gundlach, 2021).

7. Consequences, Applications, and Future Directions

Vanishing identities for rational polytopes unify and generalize a broad spectrum of phenomena in algebraic geometry, combinatorics, and mathematical physics. They are instrumental in:

• Characterizing canonical forms and adjoints of polytopes, especially in positive geometry and the amplituhedron framework.
• Enumerative geometry, including explicit criteria for Hilbert–Samuel multiplicities and singularity types along residual arrangements.
• Combinatorial technology underpinning the construction of rational Delzant polytopes and momentum polytopes in toric geometry.
• Systematic production of thin and dual-defective polytopes, critical in real and tropical enumerative geometry.
• Recursive and explicit computation of Ehrhart (quasi-)polynomials and related generating functions.

A plausible implication is that these identities serve as the algebraic backbone for classifying and producing large families of polytopes compatible with advanced geometric and combinatorial invariants, with computational advantages emerging from their finite, high-structure encoding.

The foundational role of vanishing, both local and global, is expected to inspire further research in explicit lattice polytope enumeration, canonical form theory, and applications to mathematical physics, particularly in positive geometry and scattering amplitudes. The development of these vanishing identities continues to refine and expand the interface between algebraic, combinatorial, and geometric approaches to polytope theory.