Absolute Root Separation

Polynomial Minimum Root Separation

The minimum root separation of an arbitrary polynomial P is defined as the minimum of the distances between distinct (real or complex) roots of P. Some asymptotically good lower bounds for the root separation of P are given, where P may have multiple zeros. There are applications in the analysis of complexity of algorithms and in the theory of algebraic and transcendental numbers.

Root Separation for Trinomials

We give a separation bound for the complex roots of a trinomial f ∈ 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 rat...

Separation of lentil root tip chromosomes by colchicine

Certified Complex Root Isolation via Adaptive Root Separation Bounds

We address the problem of root isolation for polynomial systems: for an affine, zero-dimensional polynomial system of N equations in N variables, we describe an algorithm to encapsulate all complex solutions into disjoint regions, each containing precisely one solution (called isolating regions). Our approach also computes the multiplicity of each solution. The main novelty is a new approach to...

Root separation for reducible integer polynomials

We construct parametric families of (monic) reducible polynomials having two roots very close to each other.