Dilates of the Hadamard Polytope

• Dilates of the Hadamard polytope are scaled versions of polytopes whose vertices are derived from Hadamard matrices, capturing intricate lattice and combinatorial structures.
• The topic utilizes quasi-polynomial expansions and Fourier-analytic methods to quantify lattice point counts and address periodic corrections in dilation processes.
• Insights from this study impact discrete geometry, algebraic complexity, and quantum information by linking combinatorial invariants with operator theory and robust entanglement constructions.

A dilate of the Hadamard polytope refers to a scaled or parametrically modified version of a polytope whose vertices are derived from a Hadamard matrix—matrices with entries $\pm 1$ and mutually orthogonal rows. The paper of these dilates is at the intersection of discrete geometry, combinatorics, algebraic complexity, Fourier analysis, tropical geometry, and operator theory. Research in this topic addresses enumeration and structure of integer/lattice points, quasi-polynomial expansions, integer decomposition properties, star configurations, spectral transforms, connections to polytopal and quantum symmetries, and implications for optimization and algebraic factorization.

1. Combinatorial Structure and Integer Point Counts

The Hadamard polytope, with vertices in $\{ -1, 1 \}^n$ (or $\{ 0, 1 \}^n$ for normalized variants), is a canonical object for studying extremal lattice point growth under dilation. When dilated by a positive integer $d$, one examines $dP = \{ dx : x \in P \}$ or, more generally, vector-dilated forms such as $P_H(b) = \{ x \in \mathbb{R}^n : Hx \leq b \}$ for a Hadamard matrix $H$. Recent work (Saraf et al., 18 Oct 2025) gives essentially tight lower bounds for the lattice point count:

• For small $d$ ($d \leq (\log n)/4$):

$|dP \cap \mathbb{Z}^n| \geq n^{\Omega(d \log n)}$

• For large $d$:

$|dP \cap \mathbb{Z}^n| \geq (d^2/4)^n$

The tightness of these bounds is corroborated by prior work on polynomial factorization in algebraic complexity, where the number of monomials in factors is limited by the number of integer points in associated polytopes (“Newton polytopes”) (Saraf et al., 18 Oct 2025). The algebraic structure reveals that every integer point $v$ in the Hadamard polytope corresponds to a uniform convex combination indexed by an affine subspace $T(v) \subseteq \mathbb{F}_2^m$:

$v = \frac{1}{|T(v)|} \sum_{c \in T(v)} h_c$

where $h_c$ are Hadamard matrix columns, and supports correspond to subspaces or their affine translates.

2. Lattice Point Enumeration: Quasi-polynomial and Piecewise Structure

When parametrizing dilates via $P_H(b)$, where $b$ varies in a cone of constant combinatorial type, the counting function

$\Phi(H, b) = \# (P_H(b) \cap \mathbb{Z}^n)$

is given by a quasi-polynomial expansion (Henk et al., 2012):

$$\Phi(H, b) = \sum_{J \in \mathcal{J}} \Phi_J(H, b) \cdot b^J, \quad b^J = b_1^{J_1} b_2^{J_2} \cdots b_k^{J_k}, \quad |J|_1 \leq n$$

where each coefficient $\Phi_J(H, b)$ is a piecewise polynomial, periodic under prescribed translations of $b$, and satisfies the differential hierarchy

$$\frac{\partial}{\partial b_l} \Phi_J(H, b) = - (J_l + 1) \cdot \Phi_{J + e_l}(H, b)$$

This result is inherited from McMullen’s theory for Minkowski-sums of rational dilates. The expansion reflects both the main volumetric contribution and detailed periodic corrections (Ehrhart-theoretic discrepancies).

3. Integer Decomposition Property and Related Invariants

The integer decomposition property (IDP) is the property that, for any $k$ and any integer point $\alpha \in kP \cap \mathbb{Z}^N$, there exist integer points $\alpha_1, ..., \alpha_k \in P \cap \mathbb{Z}^N$ such that $\alpha = \alpha_1 + ... + \alpha_k$ (Cox et al., 2012). The characterization is via a family of combinatorial invariants:

Invariant	Description	Relationship
Hva$(P)$	Minimal $k$ for very ample dilated polytope	$1 \leq$ Hva $\leq$ other invariants
$U_{\text{midp}}(P)$	Minimal $k$ for IDP to hold	Hva $\leq U_{\text{midp}}$
$H_{\text{idp}}(P)$	Minimal $k$ for all $n \geq k$: $nP$ has IDP	$U_{\text{midp}} \leq H_{\text{idp}}$
$H_{\text{Hilb}}(P)$	Max degree of minimal Hilbert basis elements	Hva $\leq H_{\text{Hilb}}$
Whole$(P)$	Max degree of “holes” (faces with no interior points)	$H_{\text{Hilb}} \leq$ Whole $\leq d-1$
$H_{\text{Ehr}}(P)$	Largest index $i$ with nonzero Ehrhart $\delta$-vector	—

These invariants are tightly related; the paper of the explicit values and inequalities for the Hadamard polytope remains an area of ongoing research.

4. Fourier-Analytic Methods and Solid Angle Sums

Fourier transform methods (Diaz et al., 2016) yield expansions for the (weighted) lattice point count in dilated polytopes. The indicator function’s transform is decomposed into contributions from faces, organized by the face poset using Stokes’ theorem:

$$\widehat{P}(\xi) = \sum_T R_T(\xi) E_T(\xi) 1_{S(T)}(\xi)$$

Poisson summation translates the spatial sum to frequency space, such that for the $t$-dilate:

$$A_P(t) = t^d \lim_{\epsilon \to 0} \sum_{\xi \in \mathbb{Z}^d} \widehat{1_P}(t\xi) e^{-\pi \epsilon \|\xi\|^2}$$

Expansion in powers of $t$ yields coefficients determined by both the polytope’s combinatorial and metric structure (facet volumes, Bernoulli polynomials), and solid angle interpretations. These methods generalize classical (Macdonald) results to real dilations and relate periodicity phenomena in Ehrhart theory and lattice enumeration.

5. Hadamard Products, Tropical Geometry, and Algebraic Connections

Hadamard products of linear spaces (coordinatewise multiplication in projective space) linearize, under tropicalization, to Minkowski sums and polytope dilations (Bocci et al., 2015). For lines $L \subset \mathbb{P}^n$ not contained in many coordinate hyperplanes, the $r^\text{th}$ Hadamard power $L^{*r}$ is a linear space of dimension $r$ (if $r \leq n$), characterized by generalized Vandermonde determinant equations. Tropical geometry allows for the calculation of expected dimensions and degrees in more general (higher-dimensional) cases, e.g.,

$$\deg(L_1*...*L_r) = \text{multinomial coefficient},$$

echoing combinatorial formulas for integer points in polytopal Minkowski sums. Connections to star configurations and extremal combinatorics further underline the relevance to the structure of Hadamard polytopes under dilation.

6. Operator Theory, Function Spaces, and Infinite-Dimensional Polytopes

In analytic function spaces, the Hadamard product acts as coefficientwise multiplication for power series; a power series $h(z)$ is a Hadamard multiplier on weighted Dirichlet spaces $\mathcal{D}_\omega$ if and only if its associated operator $T_h$ is bounded on $\ell^2$ (Mashreghi et al., 2020). The sharp norm estimate for dilates $f_r(z) = f(rz)$ with $f \in \mathcal{D}_\omega$ and $r \in [0,1)$ is:

$$\mathcal{D}_\omega(f_r) \leq r^2 (2 - r) \cdot \mathcal{D}_\omega(f)$$

The set of admissible (bounded) multipliers forms a convex body in coefficient space, analogous to a polytope in infinite dimensions (“multiplier polytope” — Editor's term). Dilation invariants and sharp inequalities encode algebraic-geometric dilation properties parallel to those of finite-dimensional polytopes.

7. Quantum Symmetries and the Birkhoff Polytope

The robust Hadamard matrices (those whose $2\times2$ principal minors are Hadamard) form foundational elements for constructing rays in the Birkhoff polytope (the polytope of bistochastic matrices). If a robust Hadamard matrix exists (guaranteed for even $n \leq 20$), then all bistochastic matrices on the ray joining any permutation matrix and the flat matrix are unistochastic (i.e., can be written as the modulus-square of a unitary) (Rajchel-Mieldzioć et al., 2018). This property extends to quantum information theory, enabling explicit construction of families of equi-entangled bases that interpolate between maximally entangled and product states.

Setting	Matrix Class	Polytope Ray	Entanglement Construction
Real robust	Skew Hadamard	Orthostochastic	Schmidt vectors X = (a, b, ..., b)
Complex robust	Conference matrix	Unistochastic	Equi-entangled Hilbert bases

These results elucidate how geometric and algebraic dilation extends to operator-theoretic and quantum settings, further connecting the combinatorics of Hadamard polytopes with applications in composite system bases.

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The paper of dilates of the Hadamard polytope thus provides a nexus for combinatorics, discrete and convex geometry, algebraic complexity, analytic operator theory, tropical geometry, and quantum information. Across these strata, the theme of “dilation”—whether by integer scaling, vector parameterization, Minkowski sum, operator norm, or quantum entanglement interpolation—manifests as structured (often extremal) growth and algebraic hierarchy, governed by precise invariants and expansion formulas. These results set tight boundaries for combinatorial enumeration, enable sensitivity and asymptotic analyses, and suggest several open questions in the direction of polytopal invariants, decomposition properties, and their functional and quantum analogues.