On the Zeros of Exponential Polynomials

We consider the problem of deciding existence real roots real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions linear differential equations focus on two problems: Zero Problem , which asks whether an polynomial has a root, and Infinite Zeros such function infinitely many roots. Our main result is that for order at most 8 decidable, subject to Schanuel’s Conjecture, whilst decidable unconditionally. show moreover decision procedure 9 would yield algorithm computing Lagrange constant any given number arbitrary precision, indicating it will be very difficult extend our decidability results higher orders.