Suppose $\lambda\_1,\lambda\_2,\cdots,\lambda\_n$ are algebraic numbers. $P\_1(t),P\_2(t),\cdots,P\_n(t)$ are polynomials with algebraic coefficients.


The question is to whether the following question is decidable.
$$\sum\_{i}^n P\_i(t)\exp(\lambda\_i t)$$
has a root $t\_0>0$.



positive root for exponential polynomial
If all numbers are real, it is decidable. This follows, for example from the general result in the paper


Vorobʹev, N. N., Jr. Deciding the consistency of a system of inequalities...


<https://link.springer.com/chapter/10.1007%2F978-1-4612-0441-1_33>.


If complex numbers are allowed it is not clear to me what the answer is.
Chebotarev has a generalization of Sturm's theorem to functions
of the form $P(x,\cos x,\sin x)$ where $P$ is a polynomial, but I do not
know a reference for a general result with complex $\lambda\_k$. I suspect it might be wrong: with complex $\lambda$ the question might be undecidable.


