Root Separation for Trinomials

TL;DR

This paper establishes a polynomial-in-logarithm separation bound for complex roots of trinomials, enabling efficient computation of the number of real roots, contrasting with the exponential bounds for general polynomials.

Contribution

It provides the first polynomial-in-sparse-encoding separation bound for trinomials and applies this to develop a polynomial-time algorithm for counting real roots.

Findings

Separation bound for trinomial roots is polynomial in the sparse encoding size.

Number of real roots of a trinomial can be computed in polynomial time.

Such bounds do not exist for 4-nomials, highlighting a unique property of trinomials.

Abstract

We give a separation bound for the complex roots of a trinomial $f \in \mathbb{Z}[X]$. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of $f$; in particular, it is polynomial in $\log (\deg f)$. It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of $f$ rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial $f$ can be computed in time polynomial in the size of the sparse encoding of~$f$. The same problem is open for 4-nomials.