# positive root for exponential polynomial

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 $$t_0>0$$.

• analytical or numerical? Sep 21 '18 at 15:56
• The question is whether there's an algorithm saying whether there's a positive root, not to compute this root.
– YCor
Sep 21 '18 at 22:40

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.