## Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing (2008)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Cheng08efficientalgorithms,

author = {Qi Cheng and Sergey P. Tarasov and Mikhail N. Vyalyi},

title = {Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing},

year = {2008}

}

### OpenURL

### Abstract

We present two deterministic polynomial time algorithms for the following problem: check whether a sparse polynomial f(x) vanishes at a given primitive nth root of unity ζn. A priori f(ζn) may be nonzero and doubly exponentially small in the input size. The existence of a polynomial time procedure in the case of factored n was conjectured by D. Plaisted in 1984, but all previously known algorithms are either randomized, or do not run in polynomial time. We apply polynomial zero testing algorithms to construct a nondeterministic polynomial time algorithm for the torsion point problem (TP). The problem TP is a particular case of the feasibility problem for a system of polynomial equations in complex numbers (coefficients of polynomials are integers). In the problem TP all coordinates of a solution must be roots of unity. Key words: algorithm, cyclotomic polynomial, root of unity, sparse representation 1

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Citation Context ...if it is known to be real. This sign determination problem of sparse real cyclotomic integers appears to be much harder than the zero testing problem. It is related to the sum of square roots problem =-=[11, 8]-=-, a famous open problem in computational geometry, which asks to determine the sign of √ a1 + · · · + √ ak − √ b1 − · · · − √ bk (28) where ai and bi are positive integers. For a prime p, the square o... |

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Citation Context ...domized argument fails. In this case coefficients of polynomials can be doubly exponentially large. The complexity of this circuit zero testing is upperbounded by a finite level of counting hierarchy =-=[1]-=- (in partiular, the problem is in PSPACE). Lower complexity bounds for this problem are unknown. Another interesting open problem is to decide whether ∑ j∈J ajζ j n is positive or negative if it is kn... |

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Citation Context ...em originally due to Chebotarev. Proposition 1. If n is a prime, then any minor of the matrix (ζ ij n )1≤i,j≤n is not zero. There are many proofs of the Chebotarev theorem. For an elementary one, see =-=[22]-=-. By studying selected minors of the matrix (ζij n )1≤i,j≤n when n is not a prime, we show that if f is a nonzero integral polynomial and all the prime factors of n are greater than sps(f), then the c... |

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Citation Context ... on vanishing sums of roots of unity. Rédei [17] and Schoenberg [19] described the lattice of coefficients of vanishing sums (see also Rédei [16], de Bruijn [4], Lam and Leung [13]). Conway and Jones =-=[7]-=- gave a lower bound on the size of the support set of a minimal vanishing sum with nonnegative coefficients. The paper by Lam and Leung [13] contains an exact characterization of the set of ℓ1-norms o... |

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Citation Context ...was studied by Plaisted [15] for the univariate case (the TP1 problem for brevity4 ) and by Rojas [18] for the multivariate case. Plaisted thus proved implicitly that TP1 is NP-hard. Koiran proved in =-=[12]-=- that FEASC ∈ AM under the Generalized Riemann Hypothesis. Of course, the same inclusion holds for the TP problem. Rojas [18] improved this result for the TP problem in various ways: TP ∈ AM under a w... |

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Citation Context ...s of running time below. 1.1 Previous work The standard algorithm for divisibility of polynomials runs in exponential time w.r.t. the input size of the problem CT. Nonetheless it is shown by Plaisted =-=[15]-=- 2that CT is in co-NP (the related problem is called SPARSE-POLY-NONROOT there). 1 Note that a linear combination of roots of unity with integer coefficients is an algebraic integer. So a straightfor... |

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Citation Context ... j mod p tk k tk−1 = jkpk + j ′ mod p tk−1 k . (11) By the Chinese remainder theorem j mod n is determined by residues modulo p tk k . Applying efficient algorithms for modular arithmetic (see, e.g., =-=[3]-=-) we get the second statement of the lemma. Note that the tk-th digit in pk-ary representation of n can be computed by the Horner scheme using O(tk) arithmetic operations. Since 2 P k tk ∏ ≤ p tk k = ... |

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Citation Context ...σ ∈ Gal(Q(ζn)/Q), then with probability at least 1/2, |σ(f(ζn))| ≥ 1/sps(f)H(f). So we can perform zero testing of sparse cyclotomic integers in randomized polynomial time. This idea has been used in =-=[5, 2]-=- to design randomized algorithm for polynomial identity testing and zero testing of expressions involving roots of rationals. Note that in some cases a large conjugate can be found deterministically. ... |

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Citation Context ...y classification results on vanishing sums of roots of unity. Rédei [17] and Schoenberg [19] described the lattice of coefficients of vanishing sums (see also Rédei [16], de Bruijn [4], Lam and Leung =-=[13]-=-). Conway and Jones [7] gave a lower bound on the size of the support set of a minimal vanishing sum with nonnegative coefficients. The paper by Lam and Leung [13] contains an exact characterization o... |

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Citation Context ...was developed by Filaseta and Schinzel (see [10, Theorem 3]). It runs in subexponential time, provided the prime power decomposition of n is given. More precisely, the estimate of the running time in =-=[10]-=- contains a factor 2s where s is the number of prime divisors of n. The algorithm is based on the observation that f(ζn) = 0 iff x n − 1 divides f(x) ∏ (x n/p − 1), also observed by Plaisted in [15]. ... |

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Citation Context ...ive coefficients. The paper by Lam and Leung [13] contains an exact characterization of the set of ℓ1-norms of vectors of the coefficients of vanishing sums with nonnegative coefficients. Steinberger =-=[21]-=- developed a method for construction of minimal sums with large coefficients. In this paper we examine the algorithmic aspects of zero testing of sums of roots of unity and consider the following prob... |

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Citation Context ....1. 1There are many classification results on vanishing sums of roots of unity. Rédei [17] and Schoenberg [19] described the lattice of coefficients of vanishing sums (see also Rédei [16], de Bruijn =-=[4]-=-, Lam and Leung [13]). Conway and Jones [7] gave a lower bound on the size of the support set of a minimal vanishing sum with nonnegative coefficients. The paper by Lam and Leung [13] contains an exac... |

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Citation Context ... is supported by the RFBR grants 08–01–00414, 05–01–02803–NTsNIL a and the grant NS 5294.2008.1. 1There are many classification results on vanishing sums of roots of unity. Rédei [17] and Schoenberg =-=[19]-=- described the lattice of coefficients of vanishing sums (see also Rédei [16], de Bruijn [4], Lam and Leung [13]). Conway and Jones [7] gave a lower bound on the size of the support set of a minimal v... |

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Citation Context ...σ ∈ Gal(Q(ζn)/Q), then with probability at least 1/2, |σ(f(ζn))| ≥ 1/sps(f)H(f). So we can perform zero testing of sparse cyclotomic integers in randomized polynomial time. This idea has been used in =-=[5, 2]-=- to design randomized algorithm for polynomial identity testing and zero testing of expressions involving roots of rationals. Note that in some cases a large conjugate can be found deterministically. ... |

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Citation Context ...91, Russia. The work is supported by the RFBR grants 08–01–00414, 05–01–02803–NTsNIL a and the grant NS 5294.2008.1. 1There are many classification results on vanishing sums of roots of unity. Rédei =-=[17]-=- and Schoenberg [19] described the lattice of coefficients of vanishing sums (see also Rédei [16], de Bruijn [4], Lam and Leung [13]). Conway and Jones [7] gave a lower bound on the size of the suppor... |

5 |
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Citation Context ...if it is known to be real. This sign determination problem of sparse real cyclotomic integers appears to be much harder than the zero testing problem. It is related to the sum of square roots problem =-=[11, 8]-=-, a famous open problem in computational geometry, which asks to determine the sign of √ a1 + · · · + √ ak − √ b1 − · · · − √ bk (28) where ai and bi are positive integers. For a prime p, the square o... |

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Citation Context ...5, Theorem 5.1] 2 . The result of Plaisted that CT ∈ co-NP mentioned above implies GCT ∈ Σ2. A more sophisticated algorithm for GCT is described in a recent paper by Filaseta, Granville, and Schinzel =-=[9]-=-. However, it uses the same subexponential cyclotomic test. So, this algorithm cannot be applied to prove that GCT ∈ NP. Another type of problem related to the cyclotomic tests are specific cases of t... |

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2 |
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Citation Context .... This specific case of the problem FEASC is called the torsion point problem (TP for brevity) 3 . It was studied by Plaisted [15] for the univariate case (the TP1 problem for brevity4 ) and by Rojas =-=[18]-=- for the multivariate case. Plaisted thus proved implicitly that TP1 is NP-hard. Koiran proved in [12] that FEASC ∈ AM under the Generalized Riemann Hypothesis. Of course, the same inclusion holds for... |

2 | probabilistic algorithms for verification of polynomial identities - Fast |

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An Efficient Algorithm for Zero-Testing of a Lacunary Polynomial at the Roots of Unity
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Citation Context ...onals. Note that in some cases a large conjugate can be found deterministically. (See Theorem 2 in [6].) To our knowledge the best deterministic zero testing algorithm prior to our extended abstracts =-=[6, 23]-=- was developed by Filaseta and Schinzel (see [10, Theorem 3]). It runs in subexponential time, provided the prime power decomposition of n is given. More precisely, the estimate of the running time in... |

1 |
and right kernels of a planar multiindex transportation problem
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Citation Context ...elated to the transportation problem and to the marginal distributions of multivariate probabilistic distributions. Cyclotomic arrays form a right kernel of a planar multiindex transportation problem =-=[24]-=-. A link to marginal distributions is based on the following fact. If two n-variate distributions p1 and p2 have the same (n − 1)-variate marginal distributions then p1 − p2 is orthogonal to a space o... |