Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

TL;DR

The paper extends trigonometric polynomial optimization to weight lattices with crystallographic symmetry by reformulating invariant problems as polynomial optimizations on a compact semi-algebraic set $\mathcal{T}$ using generalized Chebyshev polynomials. It develops a Chebyshev-adapted Lasserre hierarchy, grounded in Hol–Scherer Positivstellensatz, to obtain converging lower bounds, and provides an accompanying Maple tool to generate the SDP data. The methodology is then applied to spectral bounds for chromatic numbers of set-avoiding graphs, deriving both analytic and numerical results for symmetric polytopes and lattices, including lattice bounds for $A_{n-1}$, crosspolytopes, Voronoï cells, and the cube. While many bounds are not tight in general, the approach yields sharp results in select cases and offers a unified framework combining crystallographic symmetry with polynomial optimization to study chromatic numbers and discrete measures. The work opens avenues for further symmetry exploitation and potential extensions to more complex lattices like $E_8$.

Abstract

Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which are invariant under the associated reflection group.The invariance assumption allows us to rewrite the objective function in terms of generalized Chebyshev polynomials. The new objective function is defined on a compact basic semi-algebraic set, so that we can benefit from the rich theory of polynomial optimization.We present an algorithm to compute the minimum: Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares conditions on the objective function in the Chebyshev basis.The degree of the sums of squares is weighted, defined by the root system. Increasing the degree yields a converging Lasserre-type hierarchy of lower bounds.This builds a bridge between trigonometric and polynomial optimization, allowing us to compare with existing techniques.The chromatic number of a set avoiding graph in the Euclidean space is defined through an optimal coloring.It can be computed via a spectral bound by minimizing a trigonometric polynomial. If the to be avoided set has crystallographic symmetry, our method has a natural application.Specifically, we compute spectral bounds for the first time for boundaries of symmetric polytopes.For several cases, the problem has such a simplified form that we can give analytical proofs for sharp spectral bounds.In other cases, we certify the sharpness numerically.

Figures (18)

• Figure 1: The root system $\mathrm{A}_{2}$ in $\mathbb{R}^3/\langle [1,1,1]^t \rangle$.
• Figure 2: The root system $\mathrm{B}_{2}$ in $\mathbb{R}^2$.
• Figure 3: The root system $\mathrm{G}_{2}$ in $\mathbb{R}^3/\langle [1,1,1]^t \rangle$.
• Figure 4: The root system $\mathrm{C}_{2}$ in $\mathbb{R}^2$.
• Figure 5: The image of the generalized cosines for the irreducible root systems of rank $2$ and $3$.

Theorems & Definitions (50)

• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• Example 2.4
• Theorem 2.5
• Definition 2.6
• Proposition 2.7
• proof
• Theorem 2.9