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Trigonometric polynomials are usually defined on the lattice of integers.Weconsider the larger class of weight and root lattices with crystallographicsymmetry.This article gives a new approach to minimize trigonometricpolynomials, which are invariant under the associated reflection group.Theinvariance assumption allows us to rewrite the objective function in terms ofgeneralized Chebyshev polynomials. The new objective function is defined on acompact basic semi-algebraic set, so that we can benefit from the rich theoryof polynomial optimization.We present an algorithm to compute the minimum:Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squaresconditions on the objective function in the Chebyshev basis.The degree of thesums of squares is weighted, defined by the root system. Increasing the degreeyields a converging Lasserre-type hierarchy of lower bounds.This builds abridge between trigonometric and polynomial optimization, allowing us tocompare with existing techniques.The chromatic number of a set avoiding graphin the Euclidean space is defined through an optimal coloring.It can becomputed via a spectral bound by minimizing a trigonometric polynomial. If theto be avoided set has crystallographic symmetry, our method has a naturalapplication.Specifically, we compute spectral bounds for the first time forboundaries of symmetric polytopes.For several cases, the problem has such asimplified form that we can give analytical proofs for sharp spectral bounds.Inother cases, we certify the sharpness numerically.

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