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Detecting lacunary perfect powers and computing their roots
, 2009
"... We consider the problem of determining whether a lacunary (also called a sparse or supersparse) 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 t ..."
Abstract

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We consider the problem of determining whether a lacunary (also called a sparse or supersparse) 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 nonzero 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.
Supersparse black box rational function interpolation
 Manuscript
, 2011
"... We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the ..."
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We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet’s Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p − 1. The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of nonzero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms. The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.
The Sparsity Challenges
"... While much is written about the importance of sparse polynomials in computer algebra, much less is known about the complexity of advanced (i.e. anything more than multiplication!) algorithms for them. This is due to a variety of factors, not least the problems posed by cyclotomic polynomials. In th ..."
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While much is written about the importance of sparse polynomials in computer algebra, much less is known about the complexity of advanced (i.e. anything more than multiplication!) algorithms for them. This is due to a variety of factors, not least the problems posed by cyclotomic polynomials. In this paper we state a few of the challenges that sparse polynomials pose.