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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 ..."
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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.
Filtri Configurazioni Sito WEB On composite lacunary polynomials and the proof of a conjecture of Schinzel
, 705
"... Abstract. Let g(x) be a fixed nonconstant complex polynomial. It was conjectured by Schinzel that if g(h(x)) has boundedly many terms, then h(x) ∈ C[x] must also have boundedly many terms. Solving an older conjecture raised by Rényi and by Erdös, Schinzel had proved this in the special cases g(x) ..."
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Abstract. Let g(x) be a fixed nonconstant complex polynomial. It was conjectured by Schinzel that if g(h(x)) has boundedly many terms, then h(x) ∈ C[x] must also have boundedly many terms. Solving an older conjecture raised by Rényi and by Erdös, Schinzel had proved this in the special cases g(x) = xd; however that method does not extend to the general case. Here we prove the full Schinzel’s conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an “algorithmic ” parametric description of the general decomposition f(x) = g(h(x)), where f is a polynomial with a given number of terms and g, h are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with l terms and given coefficients is nontrivially decomposable if and only if the degreevector lies in the union of certain finitely many subgroups of Zl. Introduction. The behaviour of (complex) polynomials under the operation of composition has been studied by several authors, starting with J.F. Ritt (see [S2] for an account of the theory). Here we deal with this aspect when some of the involved polynomials are lacunary (also called sparse), i.e. the number of their terms is viewed as fixed, while the corresponding degrees (and coefficients) may vary. So, we write f(x) = a1x m1 +...+alx ml