Is there any known result about the necessary and sufficient conditions for the existence of zeros for a function $f(x)=\sum\_{n=1}^{N} a\_n e^{b\_n x}$, where $a\_n,b\_n \in \mathbb{R}\, \forall n=1,2,\cdots,N$, $a\_1,a\_N >0$, $b\_1 < b\_2 < \cdots < b\_N $ and $x \in \mathbb{R}$? 


It is known (see "Problem and Theorems in Analysis II" by Polya and Szego) that using a generalization of Descartes' rule of signs it possible to say that, named with $Z$ the number of changes of sign in the sequence of the $a\_n$ and with $Z\_0$ the number of zeros of $f(x)$, $Z-Z\_0 \geq 0$ is an even integer.


The number $Z-Z\_0$ should be even since for $x \rightarrow -\infty$, the dominant therm of $f(x)$ is $a\_1 e^{b\_1 x}>0$ and for $x \rightarrow +\infty$ the dominant term is $a\_N e^{b\_N x}>0$.


This gives an upper limit for the number of zeros, but there is any way to say "$f(x)$ *should have at least* $M$ *zeros*", with $0 < M \leq Z$?


Thanks in advance,


Nico



Zeros of a combination of exponentials
Note that we can assume wlog that $b\_n\geq 0.$ In the case they are rationals, writing $b\_n=p\_n/q$, with $p\_n\in\mathbb{N},\\ $ $q\in\mathbb{N}\_+,\\ $ and $t:=e^{x/q},\\ $ puts everything into the case of positive roots of a real polinomial, with not more, nor less generality. The book by Pólya and Szegő has a section on the location and number of positive roots of a polynomial; in any case, whatever you can say for it can clearly be translated for your exponential equation. Then, the case of real $b\_n$ can certainly be treated by approximation. 


