## Greatest Common Divisors of Polynomials Given by Straight-Line Programs (1988)

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Venue: | J. ACM |

Citations: | 51 - 17 self |

### BibTeX

@ARTICLE{Kaltofen88greatestcommon,

author = {Erich Kaltofen},

title = {Greatest Common Divisors of Polynomials Given by Straight-Line Programs},

journal = {J. ACM},

year = {1988},

volume = {35},

pages = {231--264}

}

### Years of Citing Articles

### OpenURL

### Abstract

. F Algorithms on multivariate polynomials represented by straight-line programs are developed irst it is shown that most algebraic algorithms can be probabilistically applied to data that is given by y r a straight-line computation. Testing such rational numeric data for zero, for instance, is facilitated b andom evaluations modulo random prime numbers. Then auxiliary algorithms are constructed that a determine the coefficients of a multivariate polynomial in a single variable. The first main result is an lgorithm that produces the greatest common divisor of the input polynomials, all in straight-line r a representation. The second result shows how to find a straight-line program for the reduced numerato nd denominator from one for the corresponding rational function. Both the algorithm for that conl c struction and the greatest common divisor algorithm are in random polynomial-time for the usua oefficient fields and output a straight-line program, which with controllably high probab...

### Citations

854 |
An introduction to the theory of numbers
- Hardy, Wright
- 1979
(Show Context)
Citation Context ...mputations for rational functions . H (cf. corollary 8.3). But first we shall review some needed properties of Pade approximants owever, we will not prove any of these properties and instead refer to =-=[4]-=- for an in depth discussion and the references into the literature. Let f (x ) = c +c x +c x + . . .sF [[x ]], c 0, d , e 0, b 0 1 2 2 0 e given. Going back to Frobenius 1881 a rational function p (x ... |

508 |
The complexity of computing the permanent
- Valiant
- 1979
(Show Context)
Citation Context ...z ) of x for d = 1 + (n +1) + (n +1) + . . . + (n +1) , in 2 n -1 g (x , x , . . . , x , z , . . . , z ). n +1 (n +1) 1,1 n ,n n -1 . Therefore the degree-unrestricted coefficients problem is #P-hard =-=[44]-=- The just mentioned example also shows that certain operations on straight-line programs - t most likely cannot be iterated without increasing the length of the output program exponen ially. Take, for... |

442 | Literate programming
- Knuth
- 1984
(Show Context)
Citation Context ...symptotic complexity which generate the straight-line programs corresponding to these com utations (4). The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins =-=[8]-=- and Brown [5]. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown's ... |

396 | Fast probabilistic algorithms for verification of polynomial identities
- SCHWARTZ
- 1980
(Show Context)
Citation Context ... obtained a - t probabilistic algorithm that determines the factor degree pattern, i.e. the total degrees and mul iplicities of the factors, of polynomials given by straight-line programs. Schwartz's =-=[37]-=- ] evaluation technique at random points and modulo large random pseudo-primes (see also [18 February 9, 1987 g p and 3) plays an important role in that and our new results. In the context of represen... |

296 |
Approximate formulas for some functions of prime numbers
- Rosser, Schoenfeld
- 1962
(Show Context)
Citation Context ... Π pκ >2 k/4 ≥ 2 log(Nφ(P)) = Nφ(P), we conclude that fewer than k/4 of the primes p 1 ,..., p k can be divisors of N φ(P). Now p k < k(log e k + log e log e k)<klog k, k ≥ 6, (c.f. Rosser-Schoenfeld =-=[36]-=-, (3.13)). Therefore, if we randomly pick a prime p, p < k log k < C φ(P) = (l + 5 + log log(B φ)) 2 l+5 log(B φ), (†) with probability > 3/4 this prime will certify that P is defined at φ. Wehav e th... |

239 |
Probabilistic algorithms for sparse polynomials
- Zippel
(Show Context)
Citation Context ...ason is that the content and primip tive part of the inputs can not be separated because some sparse polynomials have dense rimitive parts, cf [13],. 5. We also mention the heuristic GCD algorithm in =-=[7]-=- which may be practically a faster algorithm if the inputs have few variables. February 9, 1987 - t Our algorithm for computing the GCD as a straight-line program needs several innova ions. For one, w... |

135 |
A Fast Monte-Carlo Test for Primality
- Solovay, Strassen
- 1977
(Show Context)
Citation Context ...et C according to (+) f f(P ) f(P ) O FOR is1 , . . . , j = # 21/10 log C # D Select a random positive integer psC . Perform a probabilistic primality test on p , e f(P ) .g. Solovay's and Strassen's =-=[40]-=-, such that either p is certified composite or p is probA ably prime with chances1 -- 1/(8j ). In the latter case goto step E. t this point no prime was found, so go back to step L. r Step E (Evaluati... |

130 | Fast solution of Toeplitz systems of equations and computation of Padé approximations
- Brent, Gustavson, et al.
- 1980
(Show Context)
Citation Context ...omputations for rational functions (cf. corollary 8.3). But first we shall review some needed properties of Pad é approximants. However, we will not prove any of these properties and instead refer to =-=[4]-=- for an in depth discussion and the references into the literature. Let f (x) = c 0 + c 1 x + c 2 x 2 +⋅ ⋅ ⋅∈F[[x]], c 0 ≠ 0, d, e ≥ 0, be given. Going back to Frobenius 1881 a rational function p(x)/... |

125 |
The complexity of partial derivatives
- Baur, Strassen
- 1983
(Show Context)
Citation Context ...putable polynomials are already known, Strassen’s elimination of divisions [43] or the parallelization technique by Valiant et al [46],. for instance. Another such transformation by Baur and Strassen =-=[3]-=- allows to compute all first partial derivatives with growth in length by only a constant factor and without even the need of a degree bound. There are also known negative results in [45] that show, e... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...at takes P as input and evaluates it at given points. For division-free input programs such an algorithm has been constructed [31]. For formulas as inputs divisions do not cause additional difficulty =-=[32]-=-. However, the proof of the above corollary is tied to knowing the evaluation of the input program at a random point, and we do not know how the methods in the just mentioned papers can be used to sol... |

79 | Subresultants and reduced polynomial remainder sequences
- Collins
- 1967
(Show Context)
Citation Context ...symptotic complexity which generate the straight-line programs corresponding to these com utations (4). The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins =-=[8]-=- and Brown [5]. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown's ... |

77 |
Fast parallel computation of polynomials using few processors
- Valiant, Skyum, et al.
- 1983
(Show Context)
Citation Context ...e on the example of computing a determinant with sparse polynomial entries in F [x , . . . , x ]. If we perform Gaussian 1 n s i elimination or the asymptically faster algorithm by Bunch and Hopcroft =-=[6]-=-, certain element n F [x , . . . , x ] need to be tested for non-zero before one can divide by them. At that point, t 1 n hese elements are computed by a straight-line program and we can probabilistic... |

65 | On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors
- Brown
- 1971
(Show Context)
Citation Context ...plexity which generate the straight-line programs corresponding to these computations (§4). The GCD problem for dense multivariate polynomials was first made feasible by work of Collins [8] and Brown =-=[5]-=-. Moses and Yun [34] showed how toapply the Hensel lemma to GCD computations. Zippel [47] invented an important technique to preserve sparsity of the multivariate GCD during Brown’s interpolation sche... |

52 |
Triangular factorization and inversion by fast matrix multiplication
- Bunch, Hopcroft
- 1974
(Show Context)
Citation Context ...n principle on the example of computing a determinant with sparse polynomial entries in F[x 1 ,..., x n]. If we perform Gaussian elimination or the asymptically faster algorithm by Bunch and Hopcroft =-=[6]-=-, certain elements in F[x 1 ,..., x n]need to be tested for non-zero before one can divide by them. At that point, these elements are computed by a straight-line program and we can probabilistically t... |

46 | The Design and Analysis of Algorithms - Aho, Hopcroft, et al. - 1974 |

43 | Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
- Ibarra, Moran
- 1983
(Show Context)
Citation Context ...otice also that the last element computed by P is agai mall in size. In what follows we combat the size blow-up by a modular technique, an idea first sugb gested in Schwartz [37] and Ibarra and Moran =-=[18]-=-. A generalization of what follows to alge raic extensions of Q can be found in [11]. For completeness we shall give the proof of the L next lemma. emma 3.1: Let P = ({x , . . . , x }, V , C , {s , . ... |

31 | Efficient parallel evaluation of straight-line code and arithmetic circuits
- Miller, Ramachandran, et al.
- 1988
(Show Context)
Citation Context ...o this corollary. The question is whether there is a parallel algorithm that takes P as input and evaluates it at given points. For division-free input programs such an algorithm has been constructed =-=[31]-=-. For formulas as inputs divisions do not cause additional difficulty [32]. However, the proof of the above corollary is tied to knowing the evaluation of the input program at a random point, and we d... |

30 |
Fast computation of Gcds
- Moenck
- 1973
(Show Context)
Citation Context ... be even less likely to miss a e c non-zero leading coefficient of a remainder. Instead of the classical Euclidean algorithm w ould also have used the asymptotically faster Knuth-Sch"onhage algor=-=ithm [33]-=-. This would r shorten the length of Q asymptotically to O (l d + M (d ) log(d )) with a different bound fo 0 the cardinality of R , see algorithm Rational Numerator and Denominator in 8 for more deta... |

29 | Factorization of polynomials given by straight-line programs
- Kaltofen
- 1989
(Show Context)
Citation Context ...utable polynomials are already known, Strassen's elimiation of divisions [43] or the parallelization technique by Valiant et al [46],. for instance. d Another such transformation by Baur and Strassen =-=[3]-=- allows to compute all first partial erivatives with growth in length by only a constant factor and without even the need of a - m degree bound. There are also known negative results in [45] that show... |

28 |
Evaluating polynomials at fixed sets of points
- Aho, Steiglitz, et al.
- 1975
(Show Context)
Citation Context ...) ≤ 2 T (d/2) + γ M(d), γ aconstant, and hence is T (d) =O(log(d) M(d)). We note that for coefficient fields F of characteristic 0 this conversion can be accomplished in even O(M(d)) arithmetic steps =-=[2]-=-. The program Q to be returned is built from ˜Q by appending instructions for the conversion. Therefore len(Q) = O(lM(d)+log(d)M(d)), but since log(d) =O(l)the first term already dominates the asympto... |

25 | On computing reciprocals of power series - Kung - 1974 |

20 | DAGWOOD A System for Manipulating Polynomials Given by Straight-Line Programs
- Freeman, Imirzian, et al.
- 1988
(Show Context)
Citation Context ...easy to conceive of suitable ones, e.g. labeled directed acyclic graphs (DAG) could be used. Amore intricate data structure was used for the first implementation of our algorithms and is described in =-=[10]-=-. At this point it is convenient to define the element size of a straight-line program as el-size(P) = Notice that the actual size of P is in bits Σ v * λ ∈ X ∪ S, * ∈ {′, ′′} size(v * λ ). O(len(P) l... |

19 |
Reducibility by algebraic projections. L’Enseignement Mathématique, XXVIII:253–268
- Valiant
- 1982
(Show Context)
Citation Context ...is NP-hard [35], a restriction necessarily has to be made. One natural additional parameter to bound polynomially, other than representation size, is the total degree of the input polynomial. Valiant =-=[45]-=- calls such families of polynomials of polynomially bounded degree and straight-line computation length p-computable. Sev eral important transformations on p-computable polynomials are already known, ... |

16 |
Berechnung und programm. i
- Strassen
- 1972
(Show Context)
Citation Context ...) + gM (d ), g a constant, and 0 t hence is T (d ) = O (log(d ) M (d )). We note that for coefficient fields F of characteristic his conversion can be accomplished in even O (M (d )) arithmetic steps =-=[2]-=-. The program Q ) = to be returned is built from Q by appending instructions for the conversion. Therefore len(Q O (l M (d ) + log(d )M (d )), but since log(d ) = O (l ) the first term already dominat... |

14 |
Gcdheu: Heuristic polynomial gcd algorithm based on integer gcd computation
- Char, Geddes, et al.
- 1989
(Show Context)
Citation Context ...ason is that the content and primitive part of the inputs can not be separated because some sparse polynomials have dense primitive parts, cf [13],. §5. We also mention the heuristic GCD algorithm in =-=[7]-=- which may be practically a faster algorithm if the inputs have few variables. Our algorithm for computing the GCD as a straight-line program needs several innovations. For one, we remove the need of ... |

13 |
Computing with polynomials given by straight-line programs II; Sparse factorization
- Kaltofen
- 1985
(Show Context)
Citation Context ...ed at b finds the coefficients of that polynomial his is the interpolation problem, which can be solved classically in len(Q ) = O (d ), or a 0 0 2 symptotically faster in len(Q ) = O (M (d ) log d ) =-=[1]-=-. Notice that the algebraic RAM perg forming interpolation does not require zero-tests of field elements. Finally, we link the pro rams Q , . . . , Q , Q properly together making sure that there is no... |

13 |
On the parallel evaluation of multivariate polynomials
- Hyafil
- 1979
(Show Context)
Citation Context ... 7.1). The resolution of the numerator and denominator complexity of rational functions has an important consequence in the theory of poly-logarithmic parallel computations. First we note that Hyafil =-=[17]-=- and Valiant et al [46]. established that families of p-computable polynomials can be evaluated in parallel in polynomial size and poly-logarithmic depth. We now can apply this result to the straight-... |

12 |
Sparse complex polynomials and polynomial reducibility
- Plaisted
- 1977
(Show Context)
Citation Context ...anipulated at all. Arithmetic operations triviall ecome additional assignments and the first problem of interest is the GCD. Since for even e m univariate sparse polynomials this operation is NP-hard =-=[35]-=-, a restriction necessarily has to b ade. One natural additional parameter to bound polynomially, other than representation size, p is the total degree of the input polynomial. Valiant [45] calls such... |

10 | Uniform closure properties of p-computable functions
- Kaltofen
- 1986
(Show Context)
Citation Context ...l factorization of a polynomial, again input and outputs in straight-line representation. As referred before, that paper also contains a discussion on the sparse conversion question. We also refer to =-=[24]-=- for a detailed outline of the main results of the factoring paper. Our third and most recent article [25] discusses an approach to replacing the input degree bound d in the Polynomial GCD algorithm, ... |

8 |
Which Polynomial Representation is Best
- Stoutemyer
- 1984
(Show Context)
Citation Context ...ed at b finds the coefficients of that polynomial his is the interpolation problem, which can be solved classically in len(Q ) = O (d ), or a 0 0 2 symptotically faster in len(Q ) = O (M (d ) log d ) =-=[1]-=-. Notice that the algebraic RAM perg forming interpolation does not require zero-tests of field elements. Finally, we link the pro rams Q , . . . , Q , Q properly together making sure that there is no... |

8 |
Reducibility by algebraic projections," L'Enseignement math'ematique 28
- Valiant
- 1982
(Show Context)
Citation Context ...lexity which generate the straight-line programs corresponding to these com utations (4). The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins [8] and Brown =-=[5]-=-. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown's interpolation ... |

7 | Single-factor Hensel lifting and its application to the straight-line complexity of certain polynomials
- Kaltofen
- 1986
(Show Context)
Citation Context ..., that paper also conl tains a discussion on the sparse conversion question. We also refer to [24] for a detailed out ine of the main results of the factoring paper. Our third and most recent article =-=[25]-=- - r discusses an approach to replacing the input degree bound d in the Polynomial GCD algo ithm, for instance, by a degree bound for the output polynomial. There also a completely t a different proof... |

6 | A primer: 11 keys to new SCRATCHPAD - Jenks - 1984 |

6 |
A comparison of algorithms for the symbolic computation of Pad é approximants
- Czapor, O
- 1984
(Show Context)
Citation Context ...he program computing these elements at ψ (y ν )=a ν,2≤ν ≤n,similarly to theorem 4.2 or Step D in the Polynomial GCD algorithm. If we use the asymptotically faster Knuth-Sch önhage procedure (see also =-=[9]-=- for a full description of the algorithm) then len(Q 2) ≤ γ 1 lM(d+e)+γ 2M(d+e)log(d + e) ≤ γ 3 lM(d+e), (§) where γ 2 and γ 3 are again constants solely depending on the polynomial multiplication pro... |

6 | von zur, ‘‘Irreducibility of multivariate polynomials - Gathen - 1985 |

6 |
The analysis of algorithms, Actes du congrès international des Mathématiciens 3
- Knuth
- 1970
(Show Context)
Citation Context ...ion c 0 ≠ 0isunessential by changing the lower bound for d. The classical Euclidean algorithm gives a O((d + e) 2 ) method for computing the (d, e)-Pad é approximant. The ingenious algorithm by Knuth =-=[27]-=- that was improved by Sch önhage [37] and applied to polynomial GCDs by Moenck [33] allows to compute the triple ( f i, s i, t i)with deg( f i) ≤ d, deg( f i−1) >d,inO(M(d+e)log(d + e)) operations in ... |

6 |
Schnelle Multiplikation von Polynomen über K örpern der Charakteristik 2
- önhage, A
- 1977
(Show Context)
Citation Context ...on dominating the time for multiplying polynomials in D[x] of maximum degree d. Notice that M(d) depends on the multiplication algorithm used and the best known asymptotic result is d log d log log d =-=[38]-=-. The cardinality of a set R is denoted by card(R). All logarithms in this paper are to base 2 unless otherwise indicated.sFebruary 9, 1987 2. Straight-Line Programs and Algebraic RAMs We first presen... |

5 |
von zur and Kaltofen, E., ‘‘Factoring sparse multivariate polynomials
- Gathen
- 1985
(Show Context)
Citation Context ...at Zippel’s approach is not random polynomial-time. The reason is that the content and primitive part of the inputs can not be separated because some sparse polynomials have dense primitive parts, cf =-=[13]-=-,. §5. We also mention the heuristic GCD algorithm in [7] which may be practically a faster algorithm if the inputs have few variables. Our algorithm for computing the GCD as a straight-line program n... |

5 |
Computing with polynomials given bystraight-line programs II; Sparse factorization
- Kaltofen
- 1985
(Show Context)
Citation Context ...ial-time and requires polynomially many random bit choices. We will not state explicit polynomial upper bounds for this or any ofthe subsequent algorithms, although the original version of this paper =-=[21]-=- contained several of them. Instead we now refer to [10] for the actual performance of our algorithms, which would not be captured by those crude upper bounds. However, the theoretical failure probabi... |

4 |
Probabilistic Algorithms and Straightline Programs for some rank decision problems
- Ibarra, Moran, et al.
- 1981
(Show Context)
Citation Context ... the program computing these elements at y(y ) = a , 2snsn , similarly to theorem 4. r Step D in the Polynomial GCD algorithm. If we use the asymptotically faster KnuthSch "onhage procedure (see =-=also [9]-=- for a full description of the algorithm) then len(Q )sg l M (d +e ) + g M (d +e ) log(d +e )sg l M (d +e ), () 2 2 1 2 3 3 where g and g are again constants solely depending on the polynomial multipl... |

4 |
V.,“Vermeidung von Divisionen,” J. reine u. angew
- Strassen
- 1973
(Show Context)
Citation Context ...of polynomially bounded degree and straight-line computation length p-computable. Sev eral important transformations on p-computable polynomials are already known, Strassen’s elimination of divisions =-=[43]-=- or the parallelization technique by Valiant et al [46],. for instance. Another such transformation by Baur and Strassen [3] allows to compute all first partial derivatives with growth in length by on... |

3 |
Finite segment p-adic number systems with applications to exact computation
- Krishnamurthy, Rao, et al.
- 1975
(Show Context)
Citation Context ... the program computing these elements at y(y ) = a , 2snsn , similarly to theorem 4. r Step D in the Polynomial GCD algorithm. If we use the asymptotically faster KnuthSch "onhage procedure (see =-=also [9]-=- for a full description of the algorithm) then len(Q )sg l M (d +e ) + g M (d +e ) log(d +e )sg l M (d +e ), () 2 2 1 2 3 3 where g and g are again constants solely depending on the polynomial multipl... |

3 |
Testing polynomials which are easy to compute, Monographie 30 de l’Enseignement Mathématique
- Heintz, Schnorr
- 1982
(Show Context)
Citation Context ...mma 1): Let 0 ≠ f ∈ E[x 1 ,..., x n], E an integral domain, R ⊂ E. Then for randomly selected a 1 ,..., a n ∈ R the probability ν =1 Prob( f (a 1,..., a n) = 0) ≤ deg( f ) card(R) . (We also refer to =-=[16]-=- for an interesting characterization of a suitable set of n-tuples that distinguishes all non-zero polynomials given byshort straight-line programs from the zero polynomial.) Lemma 4.2: Let P =({x 1,.... |

3 |
Schnelle Kettenbruchentwicklungen
- önhage, A
- 1971
(Show Context)
Citation Context ...the lower bound for d. The classical Euclidean algorithm gives a O((d + e) 2 ) method for computing the (d, e)-Pad é approximant. The ingenious algorithm by Knuth [27] that was improved by Sch önhage =-=[37]-=- and applied to polynomial GCDs by Moenck [33] allows to compute the triple ( f i, s i, t i)with deg( f i) ≤ d, deg( f i−1) >d,inO(M(d+e)log(d + e)) operations in F. We now are prepared to describe th... |

2 | DAGWOOD: A system for manipulati ing polynomials given by straight-line programs - Freeman, Imirzian, et al. - 1986 |

2 |
von zur, "Parallel arithmetic computation: A survey
- Gathen
- 1986
(Show Context)
Citation Context ...or recursive procedures, and another memory in which the alges braic arithmetic is carried out, is also reflected in other models for algebraic computations uch as the parallel arithmetic networks in =-=[12]-=- or by the omnipresence of the built-in type h Integer for indexing in the Scratchpad II language [20]. Each word in address memory can old an integral address and each word in data memory can store a... |

2 |
von zur and Kaltofen, E., "Factoring sparse multivariate polynomials
- Gathen
- 1985
(Show Context)
Citation Context ...utable polynomials are already known, Strassen's elimiation of divisions [43] or the parallelization technique by Valiant et al [46],. for instance. d Another such transformation by Baur and Strassen =-=[3]-=- allows to compute all first partial erivatives with growth in length by only a constant factor and without even the need of a - m degree bound. There are also known negative results in [45] that show... |

2 |
A note on polynomials with symmetric galois group which are easy to compute," Theor
- Heintz
- 1986
(Show Context)
Citation Context ...lexity which generate the straight-line programs corresponding to these com utations (4). The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins [8] and Brown =-=[5]-=-. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown's interpolation ... |

2 |
Arithmetic in quadratic fields with unique factorization
- Kaltofen, Rolletschek
- 1985
(Show Context)
Citation Context ...e on the example of computing a determinant with sparse polynomial entries in F [x , . . . , x ]. If we perform Gaussian 1 n s i elimination or the asymptically faster algorithm by Bunch and Hopcroft =-=[6]-=-, certain element n F [x , . . . , x ] need to be tested for non-zero before one can divide by them. At that point, t 1 n hese elements are computed by a straight-line program and we can probabilistic... |

2 |
Schnelle Kettenbruchentwicklungen
- Schonhage
- 1971
(Show Context)
Citation Context ...symptotic complexity which generate the straight-line programs corresponding to these com utations (4). The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins =-=[8]-=- and Brown [5]. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown's ... |