Detecting lacunary perfect powers and computing their roots

TL;DR

This paper presents randomized and deterministic algorithms for detecting lacunary perfect powers and computing their roots in polynomial equations, with efficiency proven over various fields and demonstrated through implementation.

Contribution

It introduces new polynomial-time algorithms for identifying perfect powers and extracting roots from lacunary polynomials, including randomized and sparsity-sensitive deterministic methods.

Findings

The randomized algorithm efficiently detects perfect powers in lacunary polynomials.

Deterministic algorithms compute roots with output-sensitive and number-theoretic approaches.

Implementation confirms practical efficiency of the proposed algorithms.

Abstract

We consider solutions to the equation f = h^r for polynomials f and h and integer r > 1. Given a polynomial f in the lacunary (also called sparse or super-sparse) representation, we first show how to determine if f can be written as h^r and, if so, to find such an r. This is a Monte Carlo randomized algorithm whose cost is polynomial in the number of non-zero terms of f and in log(deg f), i.e., polynomial in the size of the lacunary representation, and it works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give two deterministic algorithms to compute the perfect root h given f and r. The first is output-sensitive (based on the sparsity of h) and works only over Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second approach to computing h, which is extremely efficient and works over both GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of these algorithms are unconditionally polynomial-time in the lacunary size of the input polynomial f. Finally, we demonstrate the efficiency of the randomized detection algorithm and the latter perfect root computation algorithm with an implementation in the C++ library NTL.