Consider the ring parametrized by $\{a_i, b_i\}$ of functions $f(t) = \sum_i a_i e^{-b_i t}$. Is this ring quadratically closed? If not, how well can I approximate the square root of an element of the ring using another element of the ring?
We can assume complex $a_i, b_i$ where $b_i$'s real component is nonnegative.
Let the ring multiplication of two elements $f_1$, $f_2$ be regular multiplication: $f(t) = f_1(t)f_2(t)$.