3
$\begingroup$

Given $n\geq 3$ distinct constants $c_1, c_2, ..., c_n \in\mathbb{C}$, I want to bound/estimate the number of positive real roots for the equation $$f(x):=\sum_{i=1}^{n}\dfrac{c_i^n}{\prod_{j\neq i}(c_i-c_j)}\exp(c_i x) = 0$$ in some given range $(0,x_{\max}]$. I am looking for analytical bounds on number of roots. Any ideas or known reference/result for such equations will be very helpful. I did find some references on zeros of exponential sums but none specifically for positive real roots.

Edit 08/10/2023: I am particularly interested in the case when the distinct $c_{i}$'s are eigenvalues of a given matrix. In other words, the $c_i$'s are a collection of distinct reals and complex conjugate pairs.

Improve this question
$\endgroup$
5
  • $\begingroup$ This is equivalent to $\sum_{i=1}^{n}\dfrac{c_i^n}{\prod_{j\neq i}(c_i-c_j)} x^{c_i} = 0$ for $x\in (1, \exp(x_\text{max})]$. I believe using something similar to a Vandermonde matrix you can prove it has at most $n$ roots, and perhaps you can get better results by somehow making Sturm's theorem work for this $\endgroup$
    – Daniel Weber
    Commented Aug 10, 2023 at 2:39
  • $\begingroup$ @CommandMaster There is no bound that depends on $n$ only. Already for $n=1$, the equation has infinitely many roots when $c_1$ is purely imaginary, hence any bound has to depend on $x_\max$ (and on the $c_i$’s). $\endgroup$
    – Emil Jeřábek
    Commented Aug 10, 2023 at 9:12
  • $\begingroup$ @EmilJeřábek I might be misunderstanding something, but for $n=1$ isn't the equation $0 = c \exp(cx) \leftrightarrow 0 = \exp(cx)$ which has no solutions? You are correct in general, $x^c = 1$ can have infinitely many solutions for example, so what I suggested can't work. If all $c$s have distinct real parts I think $n$ is a correct bound. $\endgroup$
    – Daniel Weber
    Commented Aug 10, 2023 at 11:44
  • $\begingroup$ Sorry, for $n=1$ it indeed has no solutions. So take $n=2$ and $c_2=-c_1$ purely imaginary. $\endgroup$
    – Emil Jeřábek
    Commented Aug 10, 2023 at 12:21
  • $\begingroup$ I am not sure a bound depending only on $n$ exists when some of the $c_i$'s are complex. For example, take $n=3$, $c_1=1, c_2 = +\iota, c_3 = -\iota$ where $\iota=\sqrt{-1}$. Then our equation becomes $e^{x} - \sin x + \cos x = 0$ which has no positive roots. $\endgroup$
    – Abhishek Halder
    Commented Aug 10, 2023 at 12:48

0

Reset to default

You must log in to answer this question.