We identify a common scheme in several existing algorithms addressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as iterates of a simple differential operator known as pseudo-linear map.

We consider systems of polynomial equations and inequalities in <?TeX $\mathbb {Q}[ \boldsymbol {y}][\boldsymbol {x}]$?> Math 1 where x = (x1, …, xn) and y = (y1, …, yt). The y indeterminates are considered as parameters and we assume that when specialising them generically, the set of common complex solutions, to the obtained equations, is finite.

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 , 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).