## Interpolation of polynomials given by straight-line

Citations: | 5 - 1 self |

### BibTeX

@MISC{Garg_interpolationof,

author = {Sanchit Garg and Éric Schost},

title = {Interpolation of polynomials given by straight-line},

year = {}

}

### OpenURL

### Abstract

programs

### Citations

452 | zur Gathen and - von - 2003 |

249 |
Probabilistic algorithms for sparse polynomials
- Zippel
- 1979
(Show Context)
Citation Context ...rbitrary fields. A first work (that did not use black boxes) was Grigoriev and Karpinski’s [16], for a particular polynomial (the determinant of a Tutte matrix). Other early results are due to Zippel =-=[28,29]-=- and Ben Or and Tiwari [3], with improvements by Kaltofen and Lakshman [22]. Zippel’s algorithm requires root finding in degree τ and computing τ discrete logarithms, assuming that these logarithms ar... |

159 | On fast multiplication of polynomials over arbitrary algebras
- Cantor, Kaltofen
- 1991
(Show Context)
Citation Context ..., and by MZ a function such that integers of bit-length d can be multiplied in MZ(d) bit operations. We make the usual super-linearity assumptions of [8, Chapter 8]; by the results of Cantor-Kaltofen =-=[5]-=- and Fürer [7], one can take M(d) in O(d log(d) log log(d)) and MZ(d) in O(d log(d) 2 O(log∗ (d)) ), that is, both are in O˜(d). We also need to find roots of a squarefree polynomial (written χ below)... |

96 |
A deterministic algorithm for sparse multivariate polynomial interpolation
- Ben-Or, Tiwari
- 1988
(Show Context)
Citation Context ...(that did not use black boxes) was Grigoriev and Karpinski’s [16], for a particular polynomial (the determinant of a Tutte matrix). Other early results are due to Zippel [28,29] and Ben Or and Tiwari =-=[3]-=-, with improvements by Kaltofen and Lakshman [22]. Zippel’s algorithm requires root finding in degree τ and computing τ discrete logarithms, assuming that these logarithms are bounded by d; no bound p... |

65 | Lower bounds for Diophantine approximation
- Giusti, Hägele, et al.
- 1997
(Show Context)
Citation Context ..., eschost@uwo.ca ( Éric Schost). Preprint submitted to Elsevier 1 March 2009Our motivation comes from polynomial system solving algorithms using Hensel lifting techniques, such as those initiated by =-=[15,13,14]-=-. For systems of positive dimension, the output of these algorithms is given as a straight-line program in the “parameters” of the problem. Since these methods are used in conjunction with modular tec... |

64 | When polynomial equation systems can be solved fast
- Giusti, Heintz, et al.
(Show Context)
Citation Context ..., eschost@uwo.ca ( Éric Schost). Preprint submitted to Elsevier 1 March 2009Our motivation comes from polynomial system solving algorithms using Hensel lifting techniques, such as those initiated by =-=[15,13,14]-=-. For systems of positive dimension, the output of these algorithms is given as a straight-line program in the “parameters” of the problem. Since these methods are used in conjunction with modular tec... |

59 | Straight-line programs in geometric elimination theory
- Giusti, Heintz, et al.
- 1998
(Show Context)
Citation Context ..., eschost@uwo.ca ( Éric Schost). Preprint submitted to Elsevier 1 March 2009Our motivation comes from polynomial system solving algorithms using Hensel lifting techniques, such as those initiated by =-=[15,13,14]-=-. For systems of positive dimension, the output of these algorithms is given as a straight-line program in the “parameters” of the problem. Since these methods are used in conjunction with modular tec... |

58 |
Interpolating polynomials from their values
- Zippel
- 1990
(Show Context)
Citation Context ...rbitrary fields. A first work (that did not use black boxes) was Grigoriev and Karpinski’s [16], for a particular polynomial (the determinant of a Tutte matrix). Other early results are due to Zippel =-=[28,29]-=- and Ben Or and Tiwari [3], with improvements by Kaltofen and Lakshman [22]. Zippel’s algorithm requires root finding in degree τ and computing τ discrete logarithms, assuming that these logarithms ar... |

56 | Fast parallel algorithms for sparse multivariate polynomial interpolation over
- Grigoriev, Karpinski, et al.
- 1990
(Show Context)
Citation Context ... τ in [21]. Kaltofen [19] also suggested to compute modulo primes p for which 3p − 1 is smooth, to facilitate the discrete logarithm computations. Over finite fields, Grigoriev, Karpinski and Singer =-=[17]-=- show that sparse interpolation is possible, up to computing in extensions of the base field; their algorithm has complexity O˜(n 2 τ 6 + q 2.5 ), where q is the size of the base field. In floating-po... |

53 | Greatest common divisors of polynomials given by straight-line programs
- Kaltofen
- 1988
(Show Context)
Citation Context ...ferent: this idea is used to solve systems of the form ∑ i xm i in unknowns xi and yj). − ∑ j y m j = Am Formally, our computational model is the algebraic RAM, as defined for instance by Kaltofen in =-=[20]-=-. The cost estimates count two kinds of operations: algebraic operations in the base ring and bit operations corresponding to integer manipulations, in particular flow control of the algorithm. Concre... |

49 | Faster integer multiplication
- Fürer
- 2007
(Show Context)
Citation Context ...function such that integers of bit-length d can be multiplied in MZ(d) bit operations. We make the usual super-linearity assumptions of [8, Chapter 8]; by the results of Cantor-Kaltofen [5] and Fürer =-=[7]-=-, one can take M(d) in O(d log(d) log log(d)) and MZ(d) in O(d log(d) 2 O(log∗ (d)) ), that is, both are in O˜(d). We also need to find roots of a squarefree polynomial (written χ below) of degree n i... |

46 | Improved sparse multivariate polynomial interpolation algorithms
- Kaltofen, Yagati
- 1988
(Show Context)
Citation Context ...tic early termination as in [24]. Acknowledgements. Erich Kaltofen greatly contributed to this paper through 2numerous discussions; we also wish to thank him for sending us a copy of the manuscripts =-=[21,19]-=-. Our thanks also go to Mark Giesbrecht and Dan Roche for many discussions on this topic, and to the reviewers for their comments. Financial support from NSERC and the Canada Research Chair program is... |

45 | Randomized interpolation and approximation of sparse polynomials
- Mansour
- 1995
(Show Context)
Citation Context ...crete logarithms, assuming that these logarithms are bounded by d; no bound polynomial in log(d) is known for this task for an arbitrary base field. For polynomials with integer coefficients, Mansour =-=[26]-=- and Alon and Mansour [1] obtain a deterministic bit complexity polynomial in h, n, log(d), τ, where h is an upper bound on the bit-length of the output coefficients, but their approach does not seem ... |

43 |
The Matching Problem for Bipartite Graphs with Polvno-w mially Bounded Permanents is in NC
- Grigoriev, Karpinski
- 1987
(Show Context)
Citation Context ...or this question, no algorithm is known that would have a complexity polynomial in τ, log(d) and apply over arbitrary fields. A first work (that did not use black boxes) was Grigoriev and Karpinski’s =-=[16]-=-, for a particular polynomial (the determinant of a Tutte matrix). Other early results are due to Zippel [28,29] and Ben Or and Tiwari [3], with improvements by Kaltofen and Lakshman [22]. Zippel’s al... |

33 | Symbolic-numeric sparse interpolation of multivariate polynomials
- Giesbrecht, Labahn, et al.
(Show Context)
Citation Context ...eir algorithm has complexity O˜(n 2 τ 6 + q 2.5 ), where q is the size of the base field. In floating-point arithmetic, a numerically robust extension of Ben Or and Tiwari’s algorithm is described in =-=[11]-=-. A question close to sparse interpolation is the recovery of the sparsest shift of a polynomial [10]. Giesbrecht and Roche [12] showed recently how to recover the sparsest shift using modular methods... |

25 | Early termination in sparse interpolation algorithms
- KALTOFEN, LEE
- 2003
(Show Context)
Citation Context ... Bézout’s) are available. The determination of bounds on the number of terms is the subject of more recent work such as [27]; without such bounds, one should use probabilistic early termination as in =-=[24]-=-. Acknowledgements. Erich Kaltofen greatly contributed to this paper through 2numerous discussions; we also wish to thank him for sending us a copy of the manuscripts [21,19]. Our thanks also go to M... |

18 | Modular rational sparse multivariate polynomial interpolation
- Kaltofen, Lakshman, et al.
- 1990
(Show Context)
Citation Context ...valuation points yi ∈ Z, and where p is a “lucky” prime greater than τ. However, the probabilistic aspects are not fully understood yet. Kaltofen and Lakshman [22,21] and Kaltofen, Lakshman and Wiley =-=[23]-=- present modular versions of Ben Or and Tiwari’s algorithm, with a cost quasi-linear in τ in [21]. Kaltofen [19] also suggested to compute modulo primes p for which 3p − 1 is smooth, to facilitate th... |

17 | Change of order for regular chains in positive dimension
- Dahan, Jin, et al.
(Show Context)
Citation Context ... up to 2 80 . When the monomial representation of the output is required, dense interpolation techniques are used (such as in the implementation in the Maple RegularChains package of the algorithm of =-=[6]-=-). Obviously, sparse interpolation of straight-line programs will be useful to handle the case when the output is sparse (see e.g. [27] for considerations on the sparseness of polynomials arising in e... |

16 |
Computing rational zeros of integral polynomials by p−adic expansion
- Loos
- 1983
(Show Context)
Citation Context ...need to find roots of a squarefree polynomial (written χ below) of degree n in Z[T ], knowing that χ splits into linear factors in Z[T ]. In our context, we can slightly simplify the p-adic method of =-=[25]-=-, as we will 4know a prime p (actually, several) for which χ remains squarefree modulo p. As pointed out in [22], one can find all roots of χ modulo p by fast evaluation, using O(M(p) log(p)) operati... |

13 |
Computing with polynomials given by straight-line programs II; Sparse factorization
- Kaltofen
- 1985
(Show Context)
Citation Context ...alues of A (and some of its derivatives) at some sample points. As far as we can tell, this algorithm does not work over arbitrary rings, e.g., not in characteristic less than τ. Previously, Kaltofen =-=[18]-=- gave an algorithm with a complexity polynomial in the degree bound d. Most previous results address the related question of black-box interpolation over a field. We review here some of these results,... |

11 | Algorithms for computing the sparsest shifts for polynomials via the Berlekamp/Massey algorithm
- Giesbrecht, Kaltofen, et al.
- 2002
(Show Context)
Citation Context ...-point arithmetic, a numerically robust extension of Ben Or and Tiwari’s algorithm is described in [11]. A question close to sparse interpolation is the recovery of the sparsest shift of a polynomial =-=[10]-=-. Giesbrecht and Roche [12] showed recently how to recover the sparsest shift using modular methods; even if the question and their method are distinct from ours, it is worth mentioning that similar t... |

9 | Interpolation of shifted-lacunary polynomials
- Giesbrecht, Roche
- 2010
(Show Context)
Citation Context ...cally robust extension of Ben Or and Tiwari’s algorithm is described in [11]. A question close to sparse interpolation is the recovery of the sparsest shift of a polynomial [10]. Giesbrecht and Roche =-=[12]-=- showed recently how to recover the sparsest shift using modular methods; even if the question and their method are distinct from ours, it is worth mentioning that similar techniques are used (generat... |

9 | The Newton polytope of the implicit equation
- Sturmfels, Tevelev, et al.
(Show Context)
Citation Context ...ementation in the Maple RegularChains package of the algorithm of [6]). Obviously, sparse interpolation of straight-line programs will be useful to handle the case when the output is sparse (see e.g. =-=[27]-=- for considerations on the sparseness of polynomials arising in elimination processes). We start by a result for univariate polynomials. To state it, we use the O˜( ) notation, so as to omit logarithm... |

4 | Newton-hensel interpolation lifting
- Avendano, Krick, et al.
(Show Context)
Citation Context ...polynomial in h, n, log(d), τ, where h is an upper bound on the bit-length of the output coefficients, but their approach does not seem to extend to other rings or fields. Avendaño, Krick and Pacetti =-=[2]-=- obtain a heuristic algorithm for the interpolation of polynomials in Z[X], with a complexity polynomial in h, log(d), ˜ h, p, where ˜ h is a bound on the heights of A(yi), for some suitable evaluatio... |

3 | Evaluation properties of symmetric polynomials
- Gaudry, Schost, et al.
(Show Context)
Citation Context ...e Ai by evaluating the given straight-line program for A modulo X pi − 1; all intermediate results are thus dense polynomials of degree less than pi as well (an example of a similar computation is in =-=[9]-=-). An addition modulo Xpi − 1 takes O(pi) operations in S; a multiplication modulo Xpi − 1 takes M(pi) + pi operations, since Euclidean division by Xpi − 1 uses only pi additions. Hence, the cost of a... |

2 | epsilon-discrepancy sets and their application for interpolation of sparse polynomials
- Alon, Mansour
- 1995
(Show Context)
Citation Context ... that these logarithms are bounded by d; no bound polynomial in log(d) is known for this task for an arbitrary base field. For polynomials with integer coefficients, Mansour [26] and Alon and Mansour =-=[1]-=- obtain a deterministic bit complexity polynomial in h, n, log(d), τ, where h is an upper bound on the bit-length of the output coefficients, but their approach does not seem to extend to other rings ... |