On lacunary polynomial perfect powers

Authors:
Mark Giesbrecht;Daniel S. Roche
Affiliations:
University of Waterloo, Waterloo, ON, Canada;University of Waterloo, Waterloo, ON, Canada
Venue:
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Year:
2008

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Abstract

We consider the problem of determining whether a t-sparse or lacunary polynomial f is a perfect power, that is, f=hr for some other polynomial h and positive integer r, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in the size of the lacunary representation. The algorithm works over GF(q)[x] (at least for large characteristic) and over Z[x], where the cost is also polynomial in the log of the infinity norm of f. Subject to a conjecture, we show how to find h if it exists via a kind of sparse Newton iteration, again in time polynomial in the size of the sparse representation. Finally, we demonstrate an implementation using the C++ library NTL.