## Detecting lacunary perfect powers and computing their roots (2009)

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### BibTeX

@TECHREPORT{Giesbrecht09detectinglacunary,

author = {Mark Giesbrecht and Daniel S. Roche},

title = {Detecting lacunary perfect powers and computing their roots},

institution = {},

year = {2009}

}

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### Abstract

We consider the problem of determining whether a lacunary (also called a sparse or super-sparse) polynomial f is a perfect power, that is, f = h r for some other polynomial h and r ∈ N, 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 number of non-zero terms of f, and in terms of log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (for large characteristic) and over Z[x], where the cost is also polynomial in log ‖f‖∞. We also give a Monte Carlo algorithm to find h if it exists, for which our proposed algorithm requires polynomial time in the output size, i.e., the sparsity and height of h. Conjectures of Erdös and Schinzel, and recent work of Zannier, suggest that h must be sparse. Subject to a slightly stronger conjectures we give an extremely efficient algorithm to find h via a form of sparse Newton iteration. We demonstrate the efficiency of these algorithms with an implementation using the C++ library NTL. 1.

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Citation Context ...and practice) to find h given f = h r is by a Newton iteration. This technique has also proven to be efficient in computing perfect roots of (dense) multi-precision integers (Bach and Sorenson, 1993; =-=Bernstein, 1998-=-). In summary however, we note that both these methods require approximately linear time in the degree of f, which may be exponential in the lacunary size. Newton iteration has also been applied to fi... |

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Citation Context ...thods Two well-known techniques can be applied to the problem of testing for perfect powers, and both are very efficient when f = h r is dense. We can compute the squarefree decomposition of f as in (=-=Yun, 1976-=-), and determine whether f is a perfect power by checking whether the greatest (integer) common divisor of the exponents of all nontrivial factors in the squarefree decomposition is at least 2. An eve... |

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Citation Context ...faster method (in theory and practice) to find h given f = h r is by a Newton iteration. This technique has also proven to be efficient in computing perfect roots of (dense) multi-precision integers (=-=Bach and Sorenson, 1993-=-; Bernstein, 1998). In summary however, we note that both these methods require approximately linear time in the degree of f, which may be exponential in the lacunary size. Newton iteration has also b... |

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Citation Context ...ed by Erdös (1949) on the number of terms in the square of a polynomial. Schinzel extended this work to the case of perfect powers and proved that τ(h r ) tends to infinity as τ(h) tends to infinity (=-=Schinzel, 1987-=-). Some conjectures of Schinzel suggest that τ(h) should be O(τ(f)). A recent breakthrough of Zannier (2007) show that τ(h) is bounded by a function which does not depend on deg f, but this bound is u... |

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Citation Context ... Finally, in Section 4, we present an experimental implementation of some of our algorithms in the C++ library NTL. An earlier version of some of this work was presented in the ISSAC 2008 conference (=-=Giesbrecht and Roche, 2008-=-). ∗ c We employ soft-Oh notation: for functions σ and ϕ we say σ ∈ O˜(ϕ) if σ ∈ O(ϕlog ϕ) for some constant c > 0. 32. Testing for perfect powers In this section we describe a method to determine if... |

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