Trinomials and Deterministic Complexity Limits for Real Solving

TL;DR

This paper presents a deterministic algorithm for approximately solving real roots of trinomials with polynomial time complexity, significantly improving previous exponential bounds, and relates this to solving Koiran's Trinomial Sign Problem for most inputs.

Contribution

It introduces a nearly polynomial-time deterministic algorithm for approximate root finding of trinomials and connects this to solving Koiran's Trinomial Sign Problem for most cases.

Findings

Deterministic root approximation in time log^{4+o(1)}(dH)

Solution to Koiran's Trinomial Sign Problem for most inputs

Improved complexity bounds over previous exponential results

Abstract

We detail an algorithm that -- for all but a $\frac{1}{\Omega(\log(dH))}$ fraction of $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients in $\{-H,\ldots, H\}$ -- produces an approximate root (in the sense of Smale) for each real root of $f$ in deterministic time $\log^{4+o(1)}(dH)$ in the classical Turing model. (Each approximate root is a rational with logarithmic height $O(\log(dH))$.) The best previous deterministic bit complexity bounds were exponential in $\log d$. We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree $d$ trinomial $f\in\mathbb{Z}[x]$ with coefficients in $\{-H,\ldots,H\}$, at a point $r\!\in\!\mathbb{Q}$ of logarithmic height $\log H$, in (deterministic) time $\log^{O(1)}(dH)$. We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction $1-\frac{1}{\Omega(\log(dH))}$ of the inputs $(f,r)$.