Fourier Quasicrystals on Rⁿ Preliminary Report

This paper has three aims. First, for n ≥ 1 we construct a family of real-rooted trigonometric polynomial maps P : Cⁿ ↦ 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 {kurasovsarnak}, and for $n > 1$ they overlap with Meyer's curved model sets {meyer6} and two-dimensional {meyer7} and multidimensional {meyer8} crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in {lawton}. Second, we extend the relationship between real-rootedness and amoebas, derived for $n = 1$ by Alon, Cohen and Vinzant {alon}, to the case $n > 1.$ The extension uses results in {bushuevatsikh} 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 {gelfondkhovanskii1} . Finally, we note that Olevskii and Ulanovskii [60] have proved that all FQ are divisors of real-rooted trigonometric polynomials for $n = 1$ but that the situation for $n > 1$ remains unsolved.