Zeros of systems of exponential sums and trigonometric polynomials

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of $n$ Laurent polynomials in $(\mathbb C^{\times})n$ whose Newton polytopes have generic mutual positions. An exponential change of variables gives a similar formula for exponential sums with rational frequencies. We conjecture that this formula holds for exponential sums with real frequencies. We give an integral formula which proves the existence-part of the conjectured formula not only in the complex situation but also in a very general real setting. We also prove the conjectured formula when it gives answer zero, which happens in most cases.