Kaltofen E.: Greatest common divisors of polynomials given by Straight-line Programs. J.ACM 35 No. 1 (1988) 234-264.  Home/Search   Document Details and Download   Summary   ACM   TOC   Related Articles   Check

....O(n ) 22, note added in version posted on web] to O(n 2. 6973 ) Our algorithm combines the blocked determinant algorithm with the elimination of divisions technique of [22] Our computational model is either a straight line program arithmetic circuit or an algebraic random access machine [21]. Further problems are to compute the characteristic polynomial and the adjoint matrix of A. The main idea of [22] follows [27] and for the input matrix A computes the determinant of the polynomial matrix L(z) M z(A M) where M is an integral matrix whose entries are indepent of the entries ....

E. Kaltofen. Greatest common divisors of polynomials given by straightline programs. J. ACM, 35(1):231--264, 1988.

....3.0281 ) 22, note added in version posted on web] to O(n 2. 6973 ) Our algorithm combines the blocked determinant algorithm with the elimination of divisions technique of [22] Our computational model is either a straight line program arithmetic circuit or an algebraic random access machine [21]. Further problems are to compute the characteristic polynomial and the adjoint matrix of A. The main idea of [22] follows [27] and for the input matrix A computes the determinant of the polynomial matrix L(z) M z(A M ) where M # Z nn is an integral matrix whose entries are indepent ....

E. Kaltofen. Greatest common divisors of polynomials given by straightline programs. J. ACM, 35(1):231--264, 1988.

....encore dans les polynomes finaux, ils devront etre elimin es par sp ecialisation en des valeurs appropri ees de k, mais bien sur sans donner lieu a des annulations. Ainsi, tous les algorithmes pourront etre r ealis es par des r eseaux sur l anneau k (voir aussi [Heintz Sieveking, 1981] et [Kaltofen, 1988] pour l utilisation de cette repr esentation des polynomes en calcul formel) Dans la situation affine et dans le cas d une caract eristique positive p, nous aurons aussi a envisager l extraction de racines p i emes dans k[T 1 ; T n ] Le nombre d it erations d extractions est a priori ....

....dans la repr esentation dense comme structure de donn ees. Nous aimerions signaler ici que l approche due a [Berenstein Yger, 1991] pourrait contenir la possibilit e d un codage court des quotients dans l identit e de B ezout. Notre nouvelle version du Nullstellensatz a et e conjectur ee par E. Kaltofen en 1988 (et par bien d autres sans doute) Elle fera l objet d une publication ult erieure ( Giusti Heintz Sabia, 1991] 9 3. Situation affine 3.1. Pr eliminaires Soit k un anneau int egre comme dans 1.1 et k 0 son corps des fractions. Soient f 1 ; f s des polynomes non constants de k[x 1 ; ....

E. Kaltofen, Greatest common divisors of polynomials given by straight line programs, Journal ACM 35 No 1 (1988), 231-264.

....or ef or formula representation of size n, then the degree of p is bounded by n, and if p has an slp representation of size n, then the degree of p is at most exponential in n. The idea to use slp s as a data structure for polynomials was introduced and promoted by Erich Kaltofen (see, e.g. Kal88] Though the succinctness of this way of representing polynomials compared with full or ef representation (or other so called sparse representations considered in the literature) seems to be non negligible, one of the results of this paper will be that the complexity of the sets de ned in Sect. ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the ACM, 35:231-264, 1988.

....or ef or formula representation of size n, then the degree of p is bounded by n, and if p has an slp representation of size n, then the degree of p is at most exponential in n. The idea to use slp s as a data structure for polynomials was introduced and promoted by Erich Kaltofen (see, e.g. Kal88] Though the succinctness of this way of representing polynomials compared with full or ef representation (or other so called sparse representations considered in the literature) seems to be non negligible, one of the results of this paper will be that the complexity of the sets defined in Sect. ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the ACM, 35:231--264, 1988.

....which need be Euclidean. One caveat is that although the algorithms of Schonhage and Moenck, et al. are worst case asymptotically the fastest for their respective domains, it seems that modular methods are favored in practice. Also, the use of probabilistic algorithms (see Schonhage [12] Kaltofen [4]) may be more practical. There is also an entirely different development of GCD algorithms over the ring D[x] where D is an integral domain [3] Finally, Strassen [13] has shown that the method of Schonhage is asymptotically optimal with a bound of Theta(n log n) using a different model of ....

.... if deg(b) m then return(E) 2] a 0 a div x m ; b 0 b div x m ; fnow deg(a 0 ) m 0 where m m 0 = deg(a)g R HGCD(a 0 ; b 0 ) f l m 0 2 m is the magic threshold for this recursive callg a 0 b 0 R Gamma1 a b ; 3] if deg(b 0 ) m then return(R) [4] q a 0 div b 0 ; c d b 0 a 0 mod b 0 ; 5] l deg(c) k 2m Gamma l; fnow l Gamma m l m 0 2 m g [6] c 0 c div x k ; d 0 d div x k ; fnow deg(c 0 ) 2(l Gamma m)g S HGCD(c 0 ; d 0 ) fl Gamma m is magic threshold for this recursive call. If ....

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E. Kaltofen. Greatest common divisors of polynomials given by straightline programs. Journal of the ACM, 35:231--264, 1988.

....distance between p and q is one becomes (p x Gamma q x ) 2 (p y Gamma q y ) 2 = 1. Can such groupings be exploited A recent paper of Stifter [28] indicates a promising approach. Other techniques. A very potent technique to combat intractability is randomization. See the work of Kaltofen [17, 18] and Schwartz [24] Also, the use of datastructures in algebraic computing has not been developed up to this point, and we are investigating such issues. 4 Exact Arithmetic In working with exact arithmetic over the integers or any algebraic number ring, the cost of performing arithmetic ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the ACM, 35:231--264, 1988.

....in many symbolic computations can be palliated. Recently, polynomial GCD algorithms have been developed that use implicit representations and thus avoid the computationally costly content and primitive part computations needed in those GCD algorithms for polynomials in explicit representation [Kaltofen 1988, Kaltofen and Trager 1990a, D iaz and Kaltofen 1995] 3.2 Resultants of Multivariate Systems The solvability of a set of nonlinear multivariate polynomials can be determined by the vanishing of a generalization of the Sylvester resultant of two polynomials in a single variable. We examine two ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. J. ACM, 35(1):231--264, 1988.

.... and efficiently evaluate sums and products of certain functions (like factorials see (Bachmann, Wang, and Zima 1994) It might also be interesting to observe that a CR Phi can be considered as an encoding of a straight line program which computes the value of Phi for successive integers (Kaltofen 1986). In order to be able to use CRs to expedite the evaluation of more complicated functions (like functions with polynomial subexpressions or trigonometric functions) it is necessary to generalize the definition of Chains of Recurrences to expressions whose arguments are CRs, viz. to ....

Kaltofen, E. (1986): Greatest common divisors of polynomials given by straightline programs. Journal of the ACM 35(1): 231--264.

....can be expanded and further optimized. For example, the size of the polynomial expression trees could be greatly reduced by taking advantage of the MP common format and the set of mpp representations should be extended to include straight line programs in support of black box computations [13]. It is important to extend the number of systems that have an mpp interface: we are especially interested in Saclib, Macsyma, and Maple (but this is problematic for Maple without easy access to its internals) We are also very interested in exploring what might be called adaptive communication . ....

Kaltofen, E. Greatest Common Divisors of Polynomials Given by Straight-line Programs. Journal of the ACM 35, 1 (1986), 231--264.

.... of multivariate polynomials over a large finite field; computing the nearest multivariate polynomial with factor of constant degree and complex coe#cients in polynomial time [104] Polynomial time sparse multivariate polynomial factorization algorithms by introducing the straightline program [41, 38, 33, 30, 23, 19] and the black box representations of polynomials [56, 80] Subquadratic time polynomial factorization of univariate polynomials over a finite field [81, 96] asymptotically fastest polynomial factorization algorithm over high algebraic extensions of finite fields [92] 1.2 Linear Algebra ....

....over finite fields [93] Fastest algorithm in terms of bit operations for the determinant of an integer matrix [110] 1. 3 Divisions in Algebraic Complexity Theory Polynomial length separate computation of the numerator and denominator of a rational function given by a straight line program [38]. Asymptotically fast multiplication of polynomials over a ring [63] Fast division free computation of the determinant and the characteristic polynomial of a matrix [66, 110] Integer division with remainder in residue number systems via Newton iteration [82] 1.4 Computational Number ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. J. ACM, 35(1):231--264, 1988.

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Kaltofen, E. Greatest common divisors of polynomials given by straight-line programs. J. ACM 35, 1 (1988), 231--264.

....many symbolic computations can be palliated. Recently, polynomial GCD algorithms have been developed that use implicit representations and thus avoid the computationally costly content and primitive part computations needed in those GCD algorithms for polynomials in explicit representation [DK95, Kal88, KT90] The solvability of a set of nonlinear multivariate polynomials over the field Q can be determined by the vanishing of a generalization of the Sylvester resultant of two polynomials in a single variable. Due to the special structure of the Sylvester matrix, B ezout developed a method for ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. J. ACM, 35(1):231--264, 1988.

No context found.

Kaltofen, E., "Greatest common divisors of polynomials given by straight-line programs," J. ACM, vol. 35, no. 1, pp. 231-264, 1988.

.... a field [5, 6, 25, 27] or the algebraic closure of a field [19, 85] deterministic polynomial time irreducibility testing of multivariate polynomials over a large finite field [35] ffl Polynomial time sparse multivariate polynomial factorization algorithms by introducing the straight line program [42, 39, 24, 20, 31] and the black box representations of polynomials [57, 81] ffl Subquadratic time polynomial factorization of univariate polynomials over a finite field [82, 98] asymptotically fastest polynomial factorization algorithm over high algebraic extensions of finite fields [93] 2.2 Linear Algebra ....

....linear system solver. ffl Probabilistic analysis of the Lanczos sparse linear system solver over finite fields [94] 2. 3 Divisions in Algebraic Complexity Theory ffl Polynomial length separate computation of the numerator and denominator of a rational function given by a straight line program [39]. ffl Asymptotically fast multiplication of polynomials over a ring [64] ffl Fast division free computation of the characteristic polynomial of a matrix [67] ffl Integer division with remainder in residue number systems via Newton iteration [83] 2.4 Computational Number Theory ffl Use of ....

E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. J. ACM, 35(1):231-- 264, 1988. KaLa88

.... computation DAGs goes back to (Valiant et al. 1984) the dynamic scheme discussed in the tutorial is from (Miller et al. 1986) The problem of eliminating division from computation DAGs was first solved by Strassen (1973) for those that compute degree bounded polynomials, and then by Kaltofen (1988) for the general case of degree bounded rational functions. The latter article also contains a detailed discussion of Strassen s result. Systolic greatest common divisors The linear systolic array for polynomial and integer GCDs is due to Brent and Kung (1983) see also (Yun and Zhang 1986) ....

Kaltofen, E., "Greatest common divisors of polynomials given by straight-line programs," J. ACM 35/1, pp. 231--264 (1988).

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Kaltofen E.: Greatest common divisors of polynomials given by Straight-line Programs. J.ACM 35 No. 1 (1988) 234-264.

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E. Kaltofen. Greatest Common Divisors of Polynomials Given by Straight-Line Programs. Journal of the ACM , vol. 35 (1):pp. 231-264, January 1988.

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E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231{ 264, 1988.

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E. KALTOFEN, Greatest common divisors of polynomials given by straight-line 17 programs, J. ACM, 35 (1988), pp. 231-264.

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Erich Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231--264, January 1988.

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Erich Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231--264, January 1988.

No context found.

Erich Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231--264, January 1988.

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E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. J.ACM, 35(1), pp 231--264, January 1988.

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Erich Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231-- 264, January 1988.

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