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On Euclid's Algorithm and the Theory of Subresultants

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

doi:10.1145/321662.321665
fatcat:vvtywouhxzhonbayq5duzof4uy
*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, • • • ...##
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On Euclid's algorithm and the computation of polynomial greatest common divisors

1971
*
Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC '71
*

In fact, in the multivariate ease, the maximum

doi:10.1145/800204.806288
fatcat:vvk4wmx65zccloqfnh225gxksu
*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 ...##
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On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors

1971
*
Journal of the ACM
*

In fact, in the multivariate ease, the maximum

doi:10.1145/321662.321664
fatcat:bgu3j5rnrzgqznegt6vxxskqse
*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 ...##
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Subquadratic-time factoring of polynomials over finite fields

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

doi:10.1090/s0025-5718-98-00944-2
fatcat:rpnlr6pkybc3tfoc6p2ts4okwq
*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 . ...##
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Subquadratic-time factoring of polynomials over finite fields

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

doi:10.1145/225058.225166
dblp:conf/stoc/KaltofenS95
fatcat:esygy7ocqfgfrpgaznvcpdt5yi
*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 . ...##
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Elementary algebra revisited: Randomized algorithms
[chapter]

1998
*
DIMACS Series in Discrete Mathematics and Theoretical Computer Science
*

Their formal statement was delayed partly

doi:10.1090/dimacs/043/03
dblp:conf/dimacs/CoopermanH97
fatcat:cpgy4ggg6fbk5jusolvxxr3bsi
*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*. ...##
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Computing greatest common divisors and factorizations in quadratic number fields

1989
*
Mathematics of Computation
*

The second algorithm allows us to

doi:10.1090/s0025-5718-1989-0982367-2
fatcat:n4v7s6ics5bfvi6eaiv3nzo3cy
*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*. ...##
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The singular value decomposition for polynomial systems

1995
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Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95
*

We first give an algorithm for

doi:10.1145/220346.220371
dblp:conf/issac/CorlessGTW95
fatcat:je4opjy4lrdsxgpwh76ybnqtri
*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. ...##
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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. ...##
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The Subresultant PRS Algorithm

1978
*
ACM Transactions on Mathematical Software
*

m one or more addltmnal variables The key to controlling coefficient growth without the costly

doi:10.1145/355791.355795
fatcat:vi47n45225h3hphna5mlzyfdji
*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 ...##
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Lucas pseudoprimes

1980
*
Mathematics of Computation
*

If n is a square

doi:10.1090/s0025-5718-1980-0583518-6
fatcat:ccqsbpl4wbchdffeucfnnxwbvy
*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. ...##
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Lucas Pseudoprimes

1980
*
Mathematics of Computation
*

If n is a square

doi:10.2307/2006406
fatcat:2kwhtrylm5at7nc3unxvvffnzu
*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. ...##
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Gröbner Bases Over Tate Algebras

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

doi:10.1145/3326229.3326257
dblp:conf/issac/CarusoVV19
fatcat:kd67k3kza5hrtoft3tg33zvcjm
*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. ...##
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Gröbner bases over Tate algebras
[article]

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

arXiv:1901.09574v1
fatcat:ma5mideqyfbkblorxo2wgns3gq
*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. ...##
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Quasi-GCD computations

1985
*
Journal of Complexity
*

The reduction of integer polynomial

doi:10.1016/0885-064x(85)90024-x
fatcat:ufhjyyzdqfdq3kng5aqwwzsqsy
*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. ...
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