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.

翻译：本文研究由给定有限幂级数集的线性组合所构成的幂级数的阶。形式幂级数的阶，记作$\textrm{ord}(f)$，定义为$f(x)$中系数非零的最小$x$指数。我们的第一个结果是：这些幂级数的Wronskian的阶，与这些幂级数的线性组合中出现的最大阶，在多项式因子意义下等价。这意味着(Kayal and Saha, TOCT'2012)中用于上界平方根和阶的Wronskian方法，在多项式放大意义下是最优的。我们还在其他多种场景中证明了与(Kayal and Saha, TOCT'2012)类似的上界，用于幂级数的阶。此外，我们解决了(Etessami et al., TOCT'2014)中概述的不等式检验问题的一个特例。在论文的第二部分，我们研究了平方根和问题的等式变体，该问题因(Bl\"omer, FOCS'1991)而可在多项式时间内判定。我们探讨了当输入整数以直线程序形式给出时该问题的一个自然推广。在广义黎曼猜想(GRH)的假设下，我们证明该问题可归约到所谓的“一维”变体，并识别出解决这一“一维”变体的关键数学挑战。