Suppose we have the following function $f:\mathbb{R}^{+}\mapsto \mathbb{R}$ $$f(t)=\sum_{i=1}^k P_i(t)\exp(\alpha_i t),$$ where $\alpha_i$s are all algebraic numbers and $P_i(t)$ are all polynomials with algebraic coefficients and degree less than $m$.

There are several questions that I am interested in.

a. What is the maximal number $p$ such that there is a $t_0$ and for all $r\leq p$ $$f^{(r)}(t_0)=0.$$ b. Suppose $t_1,\cdots,t_q,\cdots$ are the real roots of $f(t)=0$, is it possible to have a converging sequence of $t_q$? In other words, is it possible to have Cauchy sequence $t_i$ such that $f(t_i)=0$? If not, do we have a lower bound of the distance between different roots?

c.Is there an algorithm to decide whether there is some common real root of $f(t)$ and $f'(t)$?

  • $\begingroup$ As to (a) a trivial general remark is that $f$ solves a linear homogeneous ODE (with constant coefficients) of order $\sum_{i=1}^k\big(1+\text{deg}P_i\big)\le km$, so certainly $p<km$ otherwise $f$ is identically zero by uniqueness of the Cauchy problem. $\endgroup$ – Pietro Majer May 23 '19 at 20:34
  • $\begingroup$ @Pietro Majer Thanks! $\endgroup$ – gondolf May 28 '19 at 5:33

a) I'm not sure in general, but if $k=2$ a function of the form $A(t) \exp(t) + A(-t) \exp(-t)$ where $A$ is a polynomial of degree $m-1$ can give you $p=m+1$ if $m$ is odd, or $m+2$ if $m$ is even, with $t_0 = 0$.

b) The function $f$ is entire, so (except in the trivial case where it is identically $0$) its roots are a discrete set, and there can't be any such Cauchy sequences.

c) What do you mean by "compute"? In most cases, I would expect the roots in question (if there are any) to be transcendental numbers not expressible in "closed form". For example, this would be the case for $ f(x) = (t + \cos(t))^2$. On the other hand, standard numerical methods can be used to get arbitrarily good approximations of roots.

  • $\begingroup$ Thanks! I changed the last question into deciding the existance of common root. What is the reason of the roots distribution of entire function? $\endgroup$ – gondolf May 25 '19 at 20:51

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