On the Hardness of PosSLP

TL;DR

This paper explores the computational complexity of PosSLP, showing that a polynomial-time solution would imply unlikely complexity class collapses, and establishes the problem's difficulty through conditional lower bounds and root counting hardness.

Contribution

It provides the first non-trivial, conditionally proven lower bound for PosSLP and links its complexity to the radical of polynomials and root counting problems.

Findings

PosSLP in BPP implies NP in BPP under certain conjectures

Counting real roots of integer polynomials is #P-hard

First non-trivial lower bound for PosSLP established conditionally

Abstract

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.