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Resolvent systems of difference polynomial ideals

Xiao-Shan Gao, Chun-Ming Yuan
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
In this paper, a new theory of resolvent systems is developed for prime difference ideals and difference ideals defined by coherent and proper irreducible ascending chains.  ...  Algorithms to compute such resolvent systems are also given.  ...  In [14] , Ritt proved similar results for ordinary differential polynomial equation systems by introducing the concept of resolvents for a prime ideal.  ... 
doi:10.1145/1145768.1145790 dblp:conf/issac/GaoY06 fatcat:iw2tbepuv5dihblqxpg3spllru

Page 134 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 64, Issue 1 [page]

1948 American Mathematical Society. Transactions of the American Mathematical Society  
We return to the elimination problem and construct a form of resolvent system for prime difference ideals.  ...  The unknowns of the ideal are determined uniquely in terms of the solutions of the resolvent system.  ... 

Page 181 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 3, Issue 2 [page]

1952 American Mathematical Society. Proceedings of the American Mathematical Society  
The polynomials of Il which are free of the y; constitute a reflexive prime difference ideal A.  ...  We obtain a solution of the resolvent system from any solution of 2 by using the equation w=L.  ... 

On Extensions of Difference Fields and the Resolvents of Prime Difference Ideals

Richard M. Cohn
1952 Proceedings of the American Mathematical Society  
We apply this analogue to the theory of the resolvent system of a reflexive prime difference ideal.  ...  The polynomials of n which are free of the y, constitute a reflexive prime difference ideal A.  ... 
doi:10.2307/2032254 fatcat:dx7wmt3iofahnp53ym2zbpbsri

On extensions of difference fields and the resolvents of prime difference ideals

Richard M. Cohn
1952 Proceedings of the American Mathematical Society  
We apply this analogue to the theory of the resolvent system of a reflexive prime difference ideal.  ...  The polynomials of n which are free of the y, constitute a reflexive prime difference ideal A.  ... 
doi:10.1090/s0002-9939-1952-0047023-2 fatcat:kob6qsmkffcw5ost6bx4rsz3zi

Page 3628 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
Any regular differential ideal is radi- 12 FIELD THEORY AND POLYNOMIALS 3628 cal (Theorem 3.6). Every prime differential ideal is regular; this is using different words, a key observation of Ritt.  ...  Now let J be a radical differential ideal of F {U, Y}; a resolvent representation for J relative to U is a regular differ- ential chain C in resolvent form in w with parametric set U such that {J,aw —«  ... 

Probabilistic algorithms for computing resolvent representations of regular differential ideals

Thomas Cluzeau, Evelyne Hubert
2008 Applicable Algebra in Engineering, Communication and Computing  
In a previous article [14] , we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations.  ...  We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination.  ...  We illustrate the method with different choices of tuples for a differential system.  ... 
doi:10.1007/s00200-008-0079-8 fatcat:azvukyd7nbgbnlzbrfrd4j5bei

Metric Dimension of Hamming Graphs and Applications to Computational Biology [article]

Lucas Laird
2020 arXiv   pre-print
We can represent the domain as the roots of a polynomial system so the apparatus of Gröbner bases comes in handy to determine, whether or not a set of nodes is resolving.  ...  In this thesis, we characterize resolvability of Hamming graphs in terms of a linear system over a finite domain: a set of nodes is resolving if and only if the linear system has only a trivial solution  ...  It follows that z is a solution of the polynomial system if and only if g(z) = 0, for all g ∈ I(P ), i.e. z is a root of each polynomial in the ideal associated with the polynomials in the system.  ... 
arXiv:2007.01337v1 fatcat:j5x43zzwkjagtgu36q7pc5rkwe

Manifolds of difference polynomials

Richard M. Cohn
1948 Transactions of the American Mathematical Society  
We return to the elimination problem and construct a form of resolvent system for prime difference ideals.  ...  The unknowns of the ideal are determined uniquely in terms of the solutions of the resolvent system.  ...  We are now prepared to construct a form of resolvent for prime ideals of difference polynomials. The resolvent unknown will be a linear combination of the unknowns of the ideal.  ... 
doi:10.1090/s0002-9947-1948-0025438-x fatcat:iu6fk44xxfcp5dqb53h3f636cm

Manifolds of Difference Polynomials

Richard M. Cohn
1948 Transactions of the American Mathematical Society  
We return to the elimination problem and construct a form of resolvent system for prime difference ideals.  ...  The unknowns of the ideal are determined uniquely in terms of the solutions of the resolvent system.  ...  We are now prepared to construct a form of resolvent for prime ideals of difference polynomials. The resolvent unknown will be a linear combination of the unknowns of the ideal.  ... 
doi:10.2307/1990560 fatcat:a6xcjva4nbbihluxu7lidzl3di

Contents and Index of Volume 30

2000 Journal of symbolic computation  
-N. and Winkler, F., Special Issue on Applications of Gröbner Bases-Foreword of the Guest Editors, 339 Using Galois Ideals for Computing Relative Resolvents, 635 Van Der Kallen, W., Complexity of the Havas  ...  , 37 Proper Parametrization of Real Tubular Surfaces, 583 Reduced Gröbner Bases, Free Difference-Differential Modules and Difference-Differential Dimension Polynomials, 357 Schicho, J., Proper Parametrization  ... 
doi:10.1006/jsco.2000.0432 fatcat:wu4uezinsjdaldfpg7bswvymyu

Optimization of genetic algorithm techniques for powder structure solution

A. J. Hanson, E. Y. Cheung, S. Habershon, K. D. M. Harris
2005 Acta Crystallographica Section A Foundations of Crystallography  
This result was later generalized by Avrami [2] showing that the positions of the peaks of the infinitely resolved Patterson map are the roots of a system of polynomial equations determined from the knowledge  ...  The point missing in these analyses is the algorithm able to select the (smallest) set of the reflections such that the corresponding intensities determine a resolvent system of polynomial equations.  ...  This result was later generalized by Avrami [2] showing that the positions of the peaks of the infinitely resolved Patterson map are the roots of a system of polynomial equations determined from the  ... 
doi:10.1107/s0108767305093116 fatcat:ucrio4cwpffrrb7shhrrsyscpu

Removal of the phase problem by the atomicity assumption

S. Ciccariello, A. Cervellino
2005 Acta Crystallographica Section A Foundations of Crystallography  
This result was later generalized by Avrami [2] showing that the positions of the peaks of the infinitely resolved Patterson map are the roots of a system of polynomial equations determined from the knowledge  ...  The point missing in these analyses is the algorithm able to select the (smallest) set of the reflections such that the corresponding intensities determine a resolvent system of polynomial equations.  ...  This result was later generalized by Avrami [2] showing that the positions of the peaks of the infinitely resolved Patterson map are the roots of a system of polynomial equations determined from the  ... 
doi:10.1107/s0108767305093104 fatcat:dopkq53pczdlnajtd3cdudlv44

On the complexity of the resolvent representation of some prime differential ideals

Lisi D'Alfonso, Gabriela Jeronimo, Pablo Solernó
2006 Journal of Complexity  
We prove upper bounds on the order and degree of the polynomials involved in a resolvent representation of the prime differential ideal associated with a polynomial differential system for a particular  ...  In addition, we give a probabilistic polynomial-time algorithm for the computation of the differential Hilbert function of the ideal.  ...  of a resolvent representation of the differential ideal associated to system (1) (see Notation 5) .  ... 
doi:10.1016/j.jco.2005.10.002 fatcat:3kkh2nn2qvasndmsn72zpngrcu

A reduction of 3-SAT problem to Buchberger algorithm [article]

Maiia Bakhova
2013 arXiv   pre-print
There is a number of known NP class problems, and majority of them have been shown to be equivalent to others.  ...  In particular now it is clear that construction of a Gröbner basis (or Buchberger algorithm) must be one of equivalent problems, but there was no example.  ...  A 3-SAT problem can be resolved using Buchberger algorithm. Proof. Given a 3-SAT problem construct the associated system of polynomial equations.  ... 
arXiv:1307.3682v1 fatcat:c7xn2b7lpbc4hlxo7lycffqx2y
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