Interpolation by decomposable univariate polynomials

TL;DR

This paper investigates the problem of interpolating a polynomial that is a composition of two univariate polynomials, providing a geometric approach and polynomial-time algorithms for most cases.

Contribution

It introduces a homotopy-based method to find decomposable interpolating polynomials and analyzes their existence and computation complexity.

Findings

Decomposable interpolating polynomials exist for almost all inputs.

A homotopy method effectively solves the interpolation problem.

The algorithm runs in polynomial time relative to the geometric data.

Abstract

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and therefore d+e-1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.