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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 $t_0>0$.

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  • $\begingroup$ analytical or numerical? $\endgroup$ – user35593 Sep 21 '18 at 15:56
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    $\begingroup$ The question is whether there's an algorithm saying whether there's a positive root, not to compute this root. $\endgroup$ – YCor Sep 21 '18 at 22:40
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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.

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