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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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Cited by 6 (1 self)
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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.
On Lacunary Polynomial Perfect Powers
, 2008
"... We consider the problem of determining whether a tsparse or lacunary 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 and log deg f, i.e., polyn ..."
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Cited by 5 (1 self)
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We consider the problem of determining whether a tsparse or lacunary 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 and log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (at least for large characteristic) and over Z[x], where the cost is also polynomial in log ‖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.
Challenges in Computational Commutative Algebra
"... Abstract. In this paper we consider a number of challenges from the point of view of the CoCoA project one of whose tasks is to develop software specialized for computations in commutative algebra. Some of the challenges extend considerably beyond the boundary of commutative algebra, and are address ..."
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Cited by 2 (0 self)
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Abstract. In this paper we consider a number of challenges from the point of view of the CoCoA project one of whose tasks is to develop software specialized for computations in commutative algebra. Some of the challenges extend considerably beyond the boundary of commutative algebra, and are addressed to the computer algebra community as a whole. 1
Sparse Polynomial Decomposition Acknowledgement
, 2007
"... All the work presented here is the result of collaboration with Mark Giesbrecht. The credit for many of the new results and ideas presented is his. However, due to the fact that this paper is given in fulfillment for the requirements of the course CS 687, none of the written text is his, and Giesbre ..."
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All the work presented here is the result of collaboration with Mark Giesbrecht. The credit for many of the new results and ideas presented is his. However, due to the fact that this paper is given in fulfillment for the requirements of the course CS 687, none of the written text is his, and Giesbrecht is not listed as an author. 1
Algorithms
"... www.cs.uwaterloo.ca/˜droche/ We consider the problem of determining whether a tsparse or lacunary polynomial f is a perfect power, that is, f = hr 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 ..."
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www.cs.uwaterloo.ca/˜droche/ We consider the problem of determining whether a tsparse or lacunary polynomial f is a perfect power, that is, f = hr 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 and log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (at least for large characteristic) and over Z[x], where the cost is also polynomial in log ‖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.
General Terms
"... We present algorithms for computing factorizations and least common left multiple (LCLM) decompositions of Ore polynomials over �q (t), for a prime power q = p µ. Our algorithms are effective in �q (t)[D; σ, δ], for any automorphism σ and σderivation δ of �q (t). On input f ∈ �q (t)[D; σ, δ], the a ..."
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We present algorithms for computing factorizations and least common left multiple (LCLM) decompositions of Ore polynomials over �q (t), for a prime power q = p µ. Our algorithms are effective in �q (t)[D; σ, δ], for any automorphism σ and σderivation δ of �q (t). On input f ∈ �q (t)[D; σ, δ], the algorithms run in time polynomial in deg D(f), deg t(f), p and µ.
ABSTRACT On Lacunary Polynomial Perfect Powers
"... We consider the problem of determining whether a tsparse or lacunary 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 and log deg f, i.e., polyn ..."
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We consider the problem of determining whether a tsparse or lacunary 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 and log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (at least for large characteristic) and over Z[x], where the cost is also polynomial in log �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. 1.