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On Euclid's Algorithm and the Theory of Subresultants

W. S. Brown, J. F. Traub
1971 Journal of the ACM  
presents an elementary treatment of the theory of subresultants, and examines the relationship of the subresultants of a given pair of polynomials to their polynomial remainder sequence as determined by  ...  Both of these algorithms reduce the coefficient growth without requiring computations of coefficient GCD's, and furthermore they are identical (up to signs) whenever the PRS is normal.  ...  To compute this GCD, Euclid's algorithm constructs the integer remainder sequence a\, a 2 , • • • , a, k , where a t is the positive remainder from the division of a t _ 2 by a t _i , for i = 3, • • •  ... 
doi:10.1145/321662.321665 fatcat:vvtywouhxzhonbayq5duzof4uy

On Euclid's algorithm and the computation of polynomial greatest common divisors

W. S. Brown
1971 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC '71  
In fact, in the multivariate ease, the maximum computing time for the modular algorithm is strictly dominated by the maximum computing time for the first pseudo-division in the classical algorithm.  ...  This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm.  ...  Hence we can bound the time for a single pseudo-division by replacing l by 2hl, d~ by d, 5 by 1, and d by 2 d 2 in (69).Multiplying the result by d, which bounds the number of pseudo-divisions, and applying  ... 
doi:10.1145/800204.806288 fatcat:vvk4wmx65zccloqfnh225gxksu

On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors

W. S. Brown
1971 Journal of the ACM  
In fact, in the multivariate ease, the maximum computing time for the modular algorithm is strictly dominated by the maximum computing time for the first pseudo-division in the classical algorithm.  ...  This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm.  ...  Hence we can bound the time for a single pseudo-division by replacing l by 2hl, d~ by d, 5 by 1, and d by 2 d 2 in (69).Multiplying the result by d, which bounds the number of pseudo-divisions, and applying  ... 
doi:10.1145/321662.321664 fatcat:bgu3j5rnrzgqznegt6vxxskqse

Subquadratic-time factoring of polynomials over finite fields

Erich Kaltofen, Victor Shoup
1998 Mathematics of Computation  
The new "baby step/giant step" techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal  ...  A very different algorithm is described by Cantor and Zassenhaus [9] (see also , especially for the case where the characteristic is 2).  ...  With these pre-computations, the total cost of computing the GCD's and divisions in the inner loop amounts to O(n 1+β+o(1) ) operations in F q .  ... 
doi:10.1090/s0025-5718-98-00944-2 fatcat:rpnlr6pkybc3tfoc6p2ts4okwq

Subquadratic-time factoring of polynomials over finite fields

Erich Kaltofen, Victor Shoup
1995 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95  
The new "baby step/giant step" techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal  ...  A very different algorithm is described by Cantor and Zassenhaus [9] (see also , especially for the case where the characteristic is 2).  ...  With these pre-computations, the total cost of computing the GCD's and divisions in the inner loop amounts to O(n 1+β+o(1) ) operations in F q .  ... 
doi:10.1145/225058.225166 dblp:conf/stoc/KaltofenS95 fatcat:esygy7ocqfgfrpgaznvcpdt5yi

Elementary algebra revisited: Randomized algorithms [chapter]

Gene Cooperman, George Havas
1998 DIMACS Series in Discrete Mathematics and Theoretical Computer Science  
Their formal statement was delayed partly by the need for rigorous analysis, but more so by the need to re-think traditional approaches to elementary algorithms.  ...  We illustrate this philosophy with some basic problems in computational number theory (GCD of many integers), linear algebra (low-rank Gaussian elimination) and group theory (random subproducts for subgroup  ...  For GCD's, one wishes to find the GCD of k integers, n 1 , . . . , n k , with few GCD computations.  ... 
doi:10.1090/dimacs/043/03 dblp:conf/dimacs/CoopermanH97 fatcat:cpgy4ggg6fbk5jusolvxxr3bsi

Computing greatest common divisors and factorizations in quadratic number fields

Erich Kaltofen, Heinrich Rolletschek
1989 Mathematics of Computation  
The second algorithm allows us to compute GCD's of algebraic integers in arbitrary number fields (ideal GCD's if the class number is > 1).  ...  We extend this result by showing that there does not even exist an input in these domains for which the GCD computation becomes possible by allowing nondecreasing norms or remainders whose norms are not  ...  It is not shown, however, how one can efficiently construct these division chains and thus compute GCD's.  ... 
doi:10.1090/s0025-5718-1989-0982367-2 fatcat:n4v7s6ics5bfvi6eaiv3nzo3cy

The singular value decomposition for polynomial systems

Robert M. Corless, Patrizia M. Gianni, Barry M. Trager, Stephen M. Watt
1995 Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95  
We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby.  ...  This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known.  ...  A first approach to multivariate GCD It is often maintained that computation of GCD's is essentially a univariate problem, since we can compute multivariate GCD's by interpolation.  ... 
doi:10.1145/220346.220371 dblp:conf/issac/CorlessGTW95 fatcat:je4opjy4lrdsxgpwh76ybnqtri

Page 6965 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
Shoup [Comput. Complexity 2 (1992), no. 3, 187-224; MR 94d:12011] for constructing normal bases. The au- thors acknowledge several contributions by H. W. Lenstra, Jr., to their proofs.  ...  The output is “axiomatically” characterized by certain divisibility and gcd conditions and thus is a “natural” object.  ... 

The Subresultant PRS Algorithm

W. S. Brown
1978 ACM Transactions on Mathematical Software  
m one or more addltmnal variables The key to controlling coefficient growth without the costly computation of GCD's of coefficmnts is the fundamental theorem of subresuitants, which shows that every polynomial  ...  However, the cost of computing the content (by applying Euclid's algorithm in the coefficient domain} may be unacceptably or even proh~bltwely high, especially if the coefficients are themselves polynomials  ...  The key to controlling coefficient growth without the costly computation GCD's of coefficients is the discovery by Collins [4] that every polynomial in a PRS is proportional to some subresultant of the  ... 
doi:10.1145/355791.355795 fatcat:vi47n45225h3hphna5mlzyfdji

Lucas pseudoprimes

Robert Baillie, Samuel S. Wagstaff
1980 Mathematics of Computation  
If n is a square divisible by p, choose D so that (Z>, n) = 1 and 41 5(p). Then 415(n) = n -1.  ...  If n is in S but n is not in T , then c>(n) is divisible by q to at least the (log log n)/(2c? -2) power.  ... 
doi:10.1090/s0025-5718-1980-0583518-6 fatcat:ccqsbpl4wbchdffeucfnnxwbvy

Lucas Pseudoprimes

Robert Baillie, Samuel S. Wagstaff
1980 Mathematics of Computation  
If n is a square divisible by p, choose D so that (Z>, n) = 1 and 41 5(p). Then 415(n) = n -1.  ...  If n is in S but n is not in T , then c>(n) is divisible by q to at least the (log log n)/(2c? -2) power.  ... 
doi:10.2307/2006406 fatcat:2kwhtrylm5at7nc3unxvvffnzu

Gröbner Bases Over Tate Algebras

Xavier Caruso, Tristan Vaccon, Thibaut Verron
2019 Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19  
We prove an analogue of the Buchberger criterion in our framework and design a Buchbergerlike and a F4-like algorithm for computing Gröbner bases over Tate algebras.  ...  Over K {X; r}, Algorithm 2 works and is correct (althought we have to be careful with the normalization of gcd's in order to avoid losses of precision as much as possible).  ...  Consequently, t is divisible by some t i, j ∈ Skel T {X; r} ≥− val r ( i ) . We deduce that τ is divisible by t i, j LT ( i ) = LT (t i, j i ), as wanted.  ... 
doi:10.1145/3326229.3326257 dblp:conf/issac/CarusoVV19 fatcat:kd67k3kza5hrtoft3tg33zvcjm

Gröbner bases over Tate algebras [article]

Xavier Caruso
2019 arXiv   pre-print
We prove an analogue of the Buchberger criterion in our framework and design a Buchberger-like and a F4-like algorithm for computing Gröbner bases over Tate algebras.  ...  Over K {X; r}, Algorithm 2 works and is correct (althought we have to be careful with the normalization of gcd's in order to avoid losses of precision as much as possible).  ...  GRÖBNER BASES OVER TATE ALGEBRAS Throughout this article, we fix a field K equipped with a discrete valuation val : K → Z ⊔ {+∞}, normalized by val(K × ) = Z.  ... 
arXiv:1901.09574v1 fatcat:ma5mideqyfbkblorxo2wgns3gq

Quasi-GCD computations

Arnold Schönhage
1985 Journal of Complexity  
The reduction of integer polynomial gcd's to integer gcd's by evaluation and interpolation has been described previously; cf.  ...  INTRODUCTION In algebraic complexity theory the (sequential) time complexity of computing polynomial gcd's is quite well understood.  ... 
doi:10.1016/0885-064x(85)90024-x fatcat:ufhjyyzdqfdq3kng5aqwwzsqsy
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