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 a minor degree that $ m $.

There are several questions that interest me.

to. What is the maximum number? $ p $ such that there is a $ t_0 $ and for all $ r leq p $

$$ f ^ {(r)} (t_0) = 0. $$

second. Suppose $ t_1, cdots, t_q, cdots $ they are the true roots of $ f (t) = 0 $Is it possible to have a convergent sequence of $ t_q $? In other words, is it possible to have a Cauchy sequence? $ t_i $ such that $ f (t_i) = 0 $? If not, do we have a lower limit on the distance between different roots?

c. Is there an algorithm to calculate all the common real roots of $ f (t) $ Y $ f & # 39; (t) $?