Harold Shapiro´s conecture and summation formulas

Harold Shapiro´s conjecture formulated more than 60-years ago suggests that two exponential polynomials have infinitely many common zeroes if and only if they aremultiples of another exponential polynomial.

Despite numerous attempts to prove, the conjecture remains unsolved in full generality, only few special cases have been settled so far. Such polynomials are important in algebraic geometry and transcendental number theory.

Real-rooted exponential polynomials were recently used to construct crystalline measures and Fourier quasicrystals solving several long-standing questions in Fourier analysis, which makes it possible to classify all Fourier quasicrystals with positive integer weights.

It was recently shown that behind every real-rooted exponential polynomial there is a pair of stable multivariate polynomials invariant under involution (Lee-Yang pair).

This opens up the possibility of applying methods of algebraic geometry and Diophantine analysis to study zeroes of real-rooted exponential polynomials.

The Lee-Yang property of polynomials not only implies that they are real-rooted, but determines very special properties of their amoebae (constructed after Gelfand-Kapranov-Zelevinsky).

The goal of the research program is to combine these methods to prove Shapiro’s conjecture first for real-rooted and then for arbitrary exponential polynomials. Summation formulas connected with multidimensional Fourier quasicrystals and  hypergraphs/fractals will be discussed.