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Subresultants revisited

Joachim von zur Gathen, Thomas Lücking
2003 Theoretical Computer Science  
Subresultants and polynomial remainder sequences are an important tool in polynomial computer algebra.  ...  polynomial subresultants.  ...  The polynomial subresultant Two slightly di erent descriptions of polynomial subresultants are in the literature.  ... 
doi:10.1016/s0304-3975(02)00639-4 fatcat:kvkib37ubvct3hpsiwia4oncam

Solving Bivariate Polynomial Systems on a GPU

Marc Moreno Maza, Wei Pan
2012 Journal of Physics, Conference Series  
Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable.  ...  This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures.  ...  For instance, techniques that have appeared to be very effective for multicore implementation [18, 19] do not apply to GPU implementation and had to be revisited [17] .  ... 
doi:10.1088/1742-6596/341/1/012022 fatcat:lioq3d2zavazbk335hiqakekfy

Solving bivariate polynomial systems on a GPU

Marc Moreno Maza, Wei Pan
2011 ACM Communications in Computer Algebra  
Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable.  ...  This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures.  ...  For instance, techniques that have appeared to be very effective for multicore implementation [18, 19] do not apply to GPU implementation and had to be revisited [17] .  ... 
doi:10.1145/2016567.2016589 fatcat:n5mc3ad3z5b7tesjsuwwlaqjiy

Complexity of computation on real algebraic numbers

Marie-Françoise Roy†, Aviva Szpirglas‡
1990 Journal of symbolic computation  
Simultaneous Inequalities All algorithms are based on proposition 2.4, a result due to Sylvester (1853) revisited by Ben-Or, Kozen & Reif (1986) .  ...  The subresultant Sturm sequence of P is the generalized subresultant Sturm sequence associated to P and 1. REMARK 2.5.  ... 
doi:10.1016/s0747-7171(08)80035-1 fatcat:ergrpbvhozaxvcy375funn6ruq

Page 1807 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
Jiirgen Hausen (Oberwolfach-Walke) 2004c:13046 13P10 68w30 von zur Gathen, Joachim (D-PDRB-DM; Paderborn) ; Liicking, Thomas (D-PDRB-DM; Paderborn) Subresultants revisited.  ...  The fun- damental Theorem 5.3 explains the relation between PRSs and subresultants. It unifies many results in the literature on various types of PRSs.  ... 

Index

2004 Journal of symbolic computation  
regular sets: the unifying concept of subvariance, 271 BERSON, J. and VAN DEN ESSEN, A., An algorithm to find a coordinate's mate, 835 Canonical comprehensive Gröbner bases, 669 CHEN, F. and WANG, W., Revisiting  ...  problems in CNF, 235 Preface, 287 Preface to First order theorem proving, 1 Resolution-based decision procedures for the universal theory of some classes of distributive lattices with operators, 891 Revisiting  ... 
doi:10.1016/s0747-7171(03)00152-4 fatcat:gnptdsf7grf75aoiujuxvve46a

Chordal Graphs in Triangular Decomposition in Top-Down Style [article]

Chenqi Mou, Yang Bai, Jiahua Lai
2018 arXiv   pre-print
Then for Wang's method and a subresultant-based algorithm for triangular decomposition in top-down style and for a subresultant-based algorithm for regular decomposition in top-down style, we prove that  ...  Subresultant-based algorithm for regular decomposition revisited Similarly as in the previous sections, we first reformulate an algorithm in top-down style for regular decomposition due to Wang as Algorithm  ...  Wang's method revisited For the self-containness of this paper, Wang's method for triangular decomposition is outlined in Algorithm 1 below.  ... 
arXiv:1811.11023v1 fatcat:f2r5655c75gsboigor6pi37q4q

Algorithms for computing triangular decompositions of polynomial systems

Changbo Chen, Marc Moreno Maza
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11  
In the rest of the present paper, we revisit this intersection operation, defined below, and extend our preliminary study reported in [7] .  ...  It follows directly from the definition of subresultants. 2 We extend the definition of subresultants and principal subresultant coefficients in such a way that f and g are themselves subresultants.  ... 
doi:10.1145/1993886.1993904 dblp:conf/issac/ChenM11 fatcat:l4pki44enrawnhrvnxomxns4r4

Algorithms for computing triangular decomposition of polynomial systems

Changbo Chen, Marc Moreno Maza
2012 Journal of symbolic computation  
In the rest of the present paper, we revisit this intersection operation, defined below, and extend our preliminary study reported in [7] .  ...  It follows directly from the definition of subresultants. 2 We extend the definition of subresultants and principal subresultant coefficients in such a way that f and g are themselves subresultants.  ... 
doi:10.1016/j.jsc.2011.12.023 fatcat:gc2zppoymre67bsldpoduufc6e

Page 3567 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
Several classical algorithmic algebraic problems are revisited from a new perspective: efficiency measured in terms of how small the output size is.  ...  Fast matrix computation of subresultant polynomial remainder sequences. (English summary) Computer algebra in scientific computing (Samarkand, 2000), 1-11, Springer, Berlin, 2000.  ... 

Sylvester's double sums: An inductive proof of the general case

Teresa Krick, Agnes Szanto
2012 Journal of symbolic computation  
In 1853, Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic  ...  Collins revisited them in Collins (1967) , and some of their properties are still mysterious.  ...  Subresultant polynomials were also introduced by Sylvester in Sylvester (1853) . They became an important tool in polynomial computer algebra after G.  ... 
doi:10.1016/j.jsc.2012.01.003 fatcat:nsqhym7k3bbnnbc6faugr3anaa

Book reports

2005 Computers and Mathematics with Applications  
Applications. 24.1 GrSbner proof systems. 24.2 Petri nets. 24.3 Proving identities and analysis of algorithms. 24.4 Cyclohexane revisited. Notes. Exercises. Appendix. 25.  ...  Fast Euclidean Algorithm. 11.1 A Fast Euclidean Algorithm for polynomials. 11.2 Subresultants via Euclid's algorithm. Notes. Exercises. 12.  ... 
doi:10.1016/j.camwa.2005.02.001 fatcat:twqxvijfkrdx7gnr26gmsbuzvu

Page 1317 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
(253-271); Jinzhao Wu, First-order polynomial based theorem proving (273-294); Ziming Li, Greatest common right divisors, least common left multiples, and subresultants of Ore polynomials (297-324); Lihong  ...  geometric problem solving (175-204); Hongbo Li, Clifford algebra approaches to mechani- cal geometry theorem proving (205-229); Xiaorong Hou, Proving by examples (231-251); Xiao-Shan Gao, Search methods revisited  ... 

Algorithms for Computing Triangular Decompositions of Polynomial Systems [article]

Changbo Chen, Marc Moreno Maza
2011 arXiv   pre-print
In this paper, we revisit this intersection operation. Let R = k[x1, . . . , xn] be the ring of multivariate polynomials with coefficients in k and ordered variables x = x1 < · · · < xn.  ...  Observe that Steps (s1) and (s2) reduce essentially to computing the subresultant chain of p and t w.r.t. v.  ... 
arXiv:1104.0689v1 fatcat:ez5kkazvrrgpdjpstb5iosqorm

Revisiting Zariski Main Theorem from a constructive point of view

M.E. Alonso, T. Coquand, H. Lombardi
2014 Journal of Algebra  
Essentially we substitute this point by two dynamical arguments; one about gcd's, using subresultants, and another using our notion of strong transcendence.  ...  We use results about subresultants given in Lemma 2.20 (for the general theory of subresultants, see [3, Chapter 4]) We consider one such subresultant s 0 (x)T + c 1 (x)T −1 + · · · + c (x) assuming that  ...  all previous subresultants have been shown to be 0.  ... 
doi:10.1016/j.jalgebra.2014.02.003 fatcat:dv3f3hy6sfeifn54nnbzeltmzu
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