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Suppose we have an equation of the form $$ \sum_{j=1}^n w_j \exp (x_jt)=0 $$

for a set of (real?) coefficients $w_j$ and $x_j$ and for $n>1$. Is it in general true that there is a (complex) $t$ that solves that equation?

The reason I ask is that I'm trying to find the poles of a function of the form $$ \log \sum_{j=1}^n w_j \exp (x_jt) $$

That, in turn, arises as the solution to a certain class of problems in network theory. I'm trying to understand when the Taylor series for that function has a finite radius of convergence

k_moreno
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    I guess all the $w_j$ have to be non zero ... right ? – G. Fougeron Oct 28 '21 at 15:15
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    What are your thoughts about the problem? For instance, given a nonconstant entire function $f(z)$, do you know under what conditions the equation $f(z)=0$ has a solution? The way your question is currently stated, it's PSQ and will be closed soon. – Moishe Kohan Oct 28 '21 at 15:15
  • @g-fougeron: That's correct (or at least two of them must be). – k_moreno Oct 28 '21 at 15:18
  • @Moishe: Thanks. I tried to give some more context. To your specific question, no, I do not know the conditions under which f(z)=0. – k_moreno Oct 28 '21 at 15:23
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    Partial idea : if all the $x_j$ are positive integers, then you have a polynomial expression in $\exp(t)$. Those always have solutions, and you are left to solve equations of the type $X_i = \exp(t)$, which will also have solutions, unless $X_i = 0$, which does not happen for all $i$, from your assumptions. – G. Fougeron Oct 28 '21 at 15:28
  • @g-fougeron: That's exactly what I'm looking for! In particular, I think it solves the broader version of the problem that I stated. Specifically, factor out $\exp(x_{min}t)$, where $x_{min}=\min_j x_j$. Then clearly there's a pole through the argument that you gave. Do you see what I mean? – k_moreno Oct 28 '21 at 15:34
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    If the $x_j$ aren't all integers, I might expand your exponentials into sines and cosines, then apply https://math.stackexchange.com/questions/370996/roots-of-a-finite-fourier-series first to the cosine series and to the sine series since both must be zero for the whole series to be zero. – Eric Towers Oct 28 '21 at 15:36
  • You might be interested in https://lc18.uniud.it/slides/plenaries/paola-daquino.pdf slide 15 specifically. The jargon is too technical for me, but I think it answers your question exactly. – G. Fougeron Oct 28 '21 at 15:50
  • @g-fougeron: It seems to me like you answered the question? I.e. once the smallest (most negative) exponent is factored out? If you want to post that answer, I'd accept it. – k_moreno Oct 28 '21 at 16:24

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