2024/03/13 by Wayne Lawton, August Tsikh, Lawton, Wayne M +1
Materials Science · Mathematics · #32A27 #32A60 #52C23 #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2403.08659
openalex publication_date 2024/03/13 · openalex created_date 2024/03/15 · openalex updated_date 2026/07/28
This paper has three aims. First, for n ≥ 1 we construct a family of real-rooted trigonometric polynomial maps P : \mathbb Cn ↦ \mathbb Cn whose divisors are Fourier Quasicrystals (FQ). For n = 1 these divisors include the first nontrivial FQ with positive integer coefficients constructed by Kurasov and Sarnak [47, and for n > 1 they overlap with Meyer's curved model sets [65] and two-dimensional [66] and multidimensional [67] crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in [50].. Second, we extend the relationship between real-rootedness and amoebas, derived for n = 1 by Alon, Cohen and Vinzant [1], to the case n > 1. The extension uses results in [10] about homology of complements of amoebas of algebraic sets of codimension > 1. Third, we prove that the divisors of all uniformly generic real-rooted P are FQ. The proof uses the formula relating Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii [34]. Finally, we note that Olevskii and Ulanovskii [72] have proved that all FQ with positive integer weights are divisors of real-rooted trigonometric polynomials for n = 1 but that the situation for n > 1 remains unsolved.