  This paper focuses on the study of the order of power series that are linear
combinations of a given finite set of power series. The order of a formal power
series, known as $\textrm{ord}(f)$, is defined as the minimum exponent of $x$
that has a non-zero coefficient in $f(x)$. Our first result is that the order
of the Wronskian of these power series is equivalent up to a polynomial factor,
to the maximum order which occurs in the linear combination of these power
series. This implies that the Wronskian approach used in (Kayal and Saha,
TOCT'2012) to upper bound the order of sum of square roots is optimal up to a
polynomial blowup. We also demonstrate similar upper bounds, similar to those
of (Kayal and Saha, TOCT'2012), for the order of power series in a variety of
other scenarios. We also solve a special case of the inequality testing problem
outlined in (Etessami et al., TOCT'2014).
  In the second part of the paper, we study the equality variant of the sum of
square roots problem, which is decidable in polynomial time due to (Bl\"omer,
FOCS'1991). We investigate a natural generalization of this problem when the
input integers are given as straight line programs. Under the assumption of the
Generalized Riemann Hypothesis (GRH), we show that this problem can be reduced
to the so-called ``one dimensional'' variant. We identify the key mathematical
challenges for solving this ``one dimensional'' variant.
