The problem $\textrm{PosSLP}$ involves determining whether an integer computed by a given straight-line program is positive. This problem has attracted considerable attention within the field of computational complexity as it provides a complete characterization of the complexity associated with numerical computation. However, non-trivial lower bounds for $\textrm{PosSLP}$ remain unknown. In this paper, we demonstrate that $\textrm{PosSLP} \in \textrm{BPP}$ would imply that $\textrm{NP} \subseteq \textrm{BPP}$, under the assumption of a conjecture concerning the complexity of the radical of a polynomial proposed by Dutta, Saxena, and Sinhababu (STOC'2018). Our proof builds upon the established $\textrm{NP}$-hardness of determining if a univariate polynomial computed by an SLP has a real root, as demonstrated by Perrucci and Sabia (JDA'2005). Therefore, our lower bound for $\textrm{PosSLP}$ represents a significant advancement in understanding the complexity of this problem. It constitutes the first non-trivial lower bound for $\textrm{PosSLP}$ , albeit conditionally. Additionally, we show that counting the real roots of an integer univariate polynomial, given as input by a straight-line program, is $\#\textrm{P}$-hard.

翻译：$\textrm{PosSLP}$ 问题涉及判定由给定直线程序所计算的整数是否为正。该问题在计算复杂性领域引起了广泛关注，因为它为数值计算相关的复杂性提供了完整刻画。然而，关于 $\textrm{PosSLP}$ 的非平凡下界仍属未知。在本文中，我们证明：若 Dutta、Saxena 和 Sinhababu（STOC'2018）提出的关于多项式根式复杂性的猜想成立，则 $\textrm{PosSLP} \in \textrm{BPP}$ 将蕴含 $\textrm{NP} \subseteq \textrm{BPP}$。我们的证明基于 Perrucci 和 Sabia（JDA'2005）已建立的结论：判定由 SLP 计算的单变量多项式是否存在实根是 $\textrm{NP}$-难的。因此，本文关于 $\textrm{PosSLP}$ 的下界——尽管是条件性的——代表了理解该问题复杂性的重要进展，且是首个非平凡下界。此外，我们证明：计算由直线程序作为输入给出的整数单变量多项式的实根个数是 $\#\textrm{P}$-难的。