# On exponential polynomials

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 $$f'(t)$$?

• 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. May 23, 2019 at 20:34
• @Pietro Majer Thanks! May 28, 2019 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.