Elimination Theory in Differential and Difference Algebra
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摘要
Elimination theory is central in differential and difference algebra. The Wu-Ritt characteristic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems,the theory of differential Chow forms, and the theory of sparse differential and difference resultants.
Elimination theory is central in differential and difference algebra. The Wu-Ritt characteristic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems,the theory of differential Chow forms, and the theory of sparse differential and difference resultants.
Elimination theory is central in differential and difference algebra. The Wu-Ritt characteristic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems,the theory of differential Chow forms, and the theory of sparse differential and difference resultants.
引文
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[2]Kolchin E R,Differential Algebra and Algebraic Groups,Academic Press,New York-London,1973.
[3]Cohn R M,Difference Algebra,Interscience Publishers John Wiley&Sons,New York-LondonSydeny,1965.
[4]Wu W T,On the decision problem and the mechanization of theorem-proving in elementary geometry,Sci.Sinica,1978,21(2):159-172.
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[6]Wu W T,Basic Principles of Mechanical Theorem Proving in Elementary Geometries,Science Press,Beijing,1984;English translation,Springer,Wien,1994.
[7]Aubry P,Lazard D,and Moreno Maza M,On the theories of triangular sets,J.Symbolic Comput.,1999,28(1):105-124.
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[11]Wu W T,Mathematics Machenization,Science Press/Kluwer,Beijing,2001.
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[31]Li W and Li Y H,Difference Chow form,J.Algebra,2015,428:67-90.
[32]Li W,Partial differential chow forms and a type of partial differential chow varieties,ArXiv:1709.02358v1,2017.
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[41]Emiris I Z,On the complexity of sparse elimination,J.Complexity,1996,12(2):134-166.
[42]D’Andrea C,Macaulay style formulas for sparse resultants,Trans.Amer.Math.Soc.,2002,354(7):2595-2629.
[43]Ore O,Formale theorie der linearen differentialgleichungen(Zweiter Teil),J.Reine Angew.Math.,1932,168:233-252.
[44]Berkovich L M and Tsirulik V G,Differential resultants and some of their applications,Differ.Equations,1986,22:530-536.
[45]Zeilberger D,A holonomic systems approach to special functions identities,J.Comput.Appl.Math.,1990,32(3):321-368.
[46]Chyzak F and Salvy B,Non-commutative elimination in Ore algebras proves multivariate identities,J.Symbolic Comput.,1998,26(2):187-227.
[47]Carrà Ferro G,A resultant theory for systems of linear partial differential equations,Lie Groups Appl.,1994,1(1):47-55.
[48]Chardin M,Differential resultants and subresultants,Fundamentals of computation theory(Gosen,1991),volume 529 of Lecture Notes in Comput.Sci.,Springer,Berlin,1991,180-189.
[49]Li Z M,A subresultant theory for linear differential,linear difference and Ore polynomials,with applications,PhD thesis,Johannes Kepler University,1996.
[50]Hong H,Ore subresultant coefficients in solutions,Appl.Algebra Engrg.Comm.Comput.,2001,12(5):421-428.
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[52]Zwillinger D,Handbook of Differential Equations,Academic Press,San Diego,CA,3rd Ed.,1998.
[53]Rueda S L and Sendra J R,Linear complete differential resultants and the implicitization of linear DPPEs,J.Symbolic Comput.,2010,45(3):324-341.
[54]Carrà-Ferro G,A resultant theory for the systems of two ordinary algebraic differential equations,Appl.Algebra Engrg.Comm.Comput.,1997,8(6):539-560.
[55]Li W,Yuan C M,and Gao X S,Sparse differential resultant for Laurent differential polynomials,Found.Comput.Math.,2015,15(2):451-517.
[56]Zhang Z Y,Yuan C M,and Gao X S,Matrix formulae of differential resultant for first order generic ordinary differential polynomials,Computer Mathematics 9th Asian Symposium,ASCM2009,Fukuoka,Japan,December 14-17,2009,10th Asian symposium,ASCM 2012,Beijing,China,October 26-28,2012,Contributed papers and invited talks,Springer,Berlin,2014,479-503.
[57]Rueda S L,Linear sparse differential resultant formulas,Linear Algebra Appl.,2013,438(11):4296-4321.
[58]Li W,Yuan C M,and Gao X S.Sparse difference resultant,ISSAC 2013-Proceedings of the38th International Symposium on Symbolic and Algebraic Computation,ACM,New York,2013,275-282.
[59]Li W,Yuan C M,and Gao X S,Sparse difference resultant,J.Symbolic Comput.,2015,68(1):169-203.
[60]Yuan C M and Zhang Z Y,New bounds and efficient algorithm for sparse difference resultant,ar Xiv:1810.00057,2018.
[61]Golubitsky O,Kondratieva M,Ovchinnikov A,et al.,A bound for orders in differential Nullstellensatz,J.Algebra,2009,322(11):3852-3877.
[62]D’Alfonso L,Jeronimo G,and Solern′o P,Effective differential Nullstellensatz for ordinary DAEsystems with constant coefficients,J.Complexity,2014,30(5):588-603.
[63]Gustavson R,Kondratieva M,and Ovchinnikov A,New effective differential Nullstellensatz,Adv.Math.,2016,290:1138-1158.
[64]Ovchinnikov A,Pogudin G,and Scanlon T,Effective difference elimination and Nullstellensatz,ArXiv:1712.01412V2,2017.
[65]Ovchinnikov A,Pogudin G,and Vo T N,Bounds for elimination of unknowns in systems of differential-algebraic equations,ArXiv:1610.0422v6,2018.
[66]Yang L,Zeng Z B,and Zhang W N,Search dependency between algebraic equations:An algorithm applied to automated reasoning,Technical Report ICTP/91/6,International Center For Theoretical Physics,International Atomic Energy Agency,Miramare,Trieste,1991.
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[68]Gallo G and Mishra B,Efficient algorithms and bounds for Wu-Ritt characteristic sets,Effective Methods in Algebraic Geometry(Castiglioncello,1990),volume 94 of Progr.Math.,Birkh?user Boston,Boston,MA,1991,119-142.
[69]Wang D M,An elimination method for polynomial systems,J.Symbolic Comput.,1993,16(2):83-114.
[70]Wang D M,Decomposing polynomial systems into simple systems,J.Symbolic Comput.,1998,25(3):295-314.
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[2]Kolchin E R,Differential Algebra and Algebraic Groups,Academic Press,New York-London,1973.
[3]Cohn R M,Difference Algebra,Interscience Publishers John Wiley&Sons,New York-LondonSydeny,1965.
[4]Wu W T,On the decision problem and the mechanization of theorem-proving in elementary geometry,Sci.Sinica,1978,21(2):159-172.
[5]Wu W T,A constructive theory of differential algebraic geometry based on works of Ritt J Fwith particular applications to mechanical theorem-proving of differential geometries,Differential Geometry and Differential Equations(Shanghai,1985),volume 1255 of Lecture Notes in Math.,Springer,Berlin,1987,173-189.
[6]Wu W T,Basic Principles of Mechanical Theorem Proving in Elementary Geometries,Science Press,Beijing,1984;English translation,Springer,Wien,1994.
[7]Aubry P,Lazard D,and Moreno Maza M,On the theories of triangular sets,J.Symbolic Comput.,1999,28(1):105-124.
[8]Boulier F,Lazard D,Ollivier F,et al.,Representation for the radical of a finitely generated differential ideal,Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation,ISSAC’95,Montreal,Canada,July 10-12,1995,158-166.ACM Press,New York,NY,1995.
[9]Bouziane D,Kandri Rody A,and Ma?arouf H,Unmixed-dimensional decomposition of a finitely generated perfect differential ideal,J.Symbolic Comput.,2001,31(6):631-649.
[10]Hubert E,Factorization-free decomposition algorithms in differential algebra,J.Symbolic Comput.,2000,29(4-5):641-662.
[11]Wu W T,Mathematics Machenization,Science Press/Kluwer,Beijing,2001.
[12]Yang L,Zhang J Z,and Hou X R,Non-linear Algebraic Equations and Automated Theorem Proving,Shanghai Science and Technological Education Pub.,1996(in Chinese).
[13]Ritt J F and Doob J L,Systems of algebraic difference equations,Amer.J.Math.,1933,55(1-4):505-514.
[14]Ritt J F and Raudenbush H W,Ideal theory and algebraic difference equations,Trans.Amer.Math.Soc.,1939,46:445-452.
[15]Gao X S,Luo Y,and Yuan C M,A characteristic set method for ordinary difference polynomial systems,J.Symbolic Comput.,2009,44(3):242-260.
[16]Gao X S and Yuan C M,Resolvent systems of difference polynomial ideals,ISSAC 2006,ACM,New York,2006,100-108.
[17]Gao X S,Yuan C M,and Zhang G L,Ritt-Wu’s characteristic set method for ordinary difference polynomial systems with arbitrary ordering,Acta Math.Sci.Ser.B(Engl.Ed.),2009,29(4):1063-1080.
[18]Zhang G L and Gao X S.Properties of ascending chains for partial difference polynomial systems,Computer Mathematics 8th Asian Symposium,ASCM 2007,Singapore,December 15-17,2007.Revised and invited papers,307-321.Springer,Berlin,2008.
[19]Kondratieva M V,Levin A B,Mikhalev A V,et al.,Differential and Difference Dimension Polynomials,volume 461 of Mathematics and Its Applications,Kluwer Academic Publishers,Dordrecht,1999.
[20]Gao X S,Van der Hoeven J,Yuan C M,et al.,Characteristic set method for differential-difference polynomial systems,J.Symbolic Comput.,2009,44(9):1137-1163.
[21]Gel’fand I M,Kapranov M M,and Zelevinsky A V,Discriminants,Resultants,and Multidimensional Determinants,Mathematics:Theory&Applications,Birkh?user Boston,Inc.,Boston,MA,1994.
[22]Hodge W V D and Pedoe D,Methods of Algebraic Geometry,Vol.II,Cambridge Mathematical Library,Cambridge University Press,Cambridge,1994.
[23]Brownawell W D,Bounds for the degrees in the nullstellensatz,Ann.Math.,1987,126(3):577-591.
[24]Sturmfels B,Sparse elimination theory,Computational Algebraic Geometry and Commutative Algebra(Cortona,1991),Sympos.Math.,XXXIV,Cambridge University Press,Cambridge,1993,264-298.
[25]Nesterenko Y V,Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers,Izv.Akad.Nauk SSSR Ser.Mat.,1977,41:253-284.
[26]Philippon P,Critères pour l’indpendance algbrique,Inst.Hautes`Etudes Sci.Publ.Math.,1986,64:5-52.
[27]Jeronimo G,Krick T,Sabia J,et al.,The computational complexity of the Chow form,Found.Comput.Math.,2004,4(1):41-117.
[28]Gao X S,Li W,and Yuan C M,Intersection theory in differential algebraic geometry:Generic intersections and the differential Chow form,Trans.Amer.Math.Soc.,2013,365(9):4575-4632.
[29]Li W and Gao X S,Chow form for projective differential variety,J.Algebra,2012,370:344-360.
[30]Freitag J,Li W,and Scanlon T,Differential chow varieties exist,J.Lond.Math.Soc.,2017,95(1):128-156.
[31]Li W and Li Y H,Difference Chow form,J.Algebra,2015,428:67-90.
[32]Li W,Partial differential chow forms and a type of partial differential chow varieties,ArXiv:1709.02358v1,2017.
[33]Macaulay F S,The Algebraic Theory of Modular Systems,Cambridge University Press,Cambridge,1994.
[34]Eisenbud D,Schreyer F,and Weyman J,Resultants and Chow forms via exterior syzygies,J.Amer.Math.Soc.,2003,16(3):537-579.
[35]Jouanolou J P,Le formalisme du résultant,Adv.Math.,1991,90(2):117-263.
[36]Canny J,Generalised characteristic polynomials,J.Symb.Comput.,1990,9(3):241-250.
[37]Emiris I Z and Canny J F,Efficient incremental algorithms for the sparse resultant and the mixed volume,J.Symb.Comput.,1995,20(2):117-149.
[38]Emiris I Z and Pan V Y,Improved algorithms for computing determinants and resultants,J.Complexity,2005,21(1):43-71.
[39]Bernstein D N,The number of roots of a system of equations,Functional Anal.Appl.,1975,9(3):183-185.
[40]Sturmfels B,On the Newton polytope of the resultant,J.Algebraic Combin.,1994,3(2):207-236.
[41]Emiris I Z,On the complexity of sparse elimination,J.Complexity,1996,12(2):134-166.
[42]D’Andrea C,Macaulay style formulas for sparse resultants,Trans.Amer.Math.Soc.,2002,354(7):2595-2629.
[43]Ore O,Formale theorie der linearen differentialgleichungen(Zweiter Teil),J.Reine Angew.Math.,1932,168:233-252.
[44]Berkovich L M and Tsirulik V G,Differential resultants and some of their applications,Differ.Equations,1986,22:530-536.
[45]Zeilberger D,A holonomic systems approach to special functions identities,J.Comput.Appl.Math.,1990,32(3):321-368.
[46]Chyzak F and Salvy B,Non-commutative elimination in Ore algebras proves multivariate identities,J.Symbolic Comput.,1998,26(2):187-227.
[47]Carrà Ferro G,A resultant theory for systems of linear partial differential equations,Lie Groups Appl.,1994,1(1):47-55.
[48]Chardin M,Differential resultants and subresultants,Fundamentals of computation theory(Gosen,1991),volume 529 of Lecture Notes in Comput.Sci.,Springer,Berlin,1991,180-189.
[49]Li Z M,A subresultant theory for linear differential,linear difference and Ore polynomials,with applications,PhD thesis,Johannes Kepler University,1996.
[50]Hong H,Ore subresultant coefficients in solutions,Appl.Algebra Engrg.Comm.Comput.,2001,12(5):421-428.
[51]Ritt J F,Differential Equations from the Algebraic Standpoint,American Mathematical Society,New York,1932.
[52]Zwillinger D,Handbook of Differential Equations,Academic Press,San Diego,CA,3rd Ed.,1998.
[53]Rueda S L and Sendra J R,Linear complete differential resultants and the implicitization of linear DPPEs,J.Symbolic Comput.,2010,45(3):324-341.
[54]Carrà-Ferro G,A resultant theory for the systems of two ordinary algebraic differential equations,Appl.Algebra Engrg.Comm.Comput.,1997,8(6):539-560.
[55]Li W,Yuan C M,and Gao X S,Sparse differential resultant for Laurent differential polynomials,Found.Comput.Math.,2015,15(2):451-517.
[56]Zhang Z Y,Yuan C M,and Gao X S,Matrix formulae of differential resultant for first order generic ordinary differential polynomials,Computer Mathematics 9th Asian Symposium,ASCM2009,Fukuoka,Japan,December 14-17,2009,10th Asian symposium,ASCM 2012,Beijing,China,October 26-28,2012,Contributed papers and invited talks,Springer,Berlin,2014,479-503.
[57]Rueda S L,Linear sparse differential resultant formulas,Linear Algebra Appl.,2013,438(11):4296-4321.
[58]Li W,Yuan C M,and Gao X S.Sparse difference resultant,ISSAC 2013-Proceedings of the38th International Symposium on Symbolic and Algebraic Computation,ACM,New York,2013,275-282.
[59]Li W,Yuan C M,and Gao X S,Sparse difference resultant,J.Symbolic Comput.,2015,68(1):169-203.
[60]Yuan C M and Zhang Z Y,New bounds and efficient algorithm for sparse difference resultant,ar Xiv:1810.00057,2018.
[61]Golubitsky O,Kondratieva M,Ovchinnikov A,et al.,A bound for orders in differential Nullstellensatz,J.Algebra,2009,322(11):3852-3877.
[62]D’Alfonso L,Jeronimo G,and Solern′o P,Effective differential Nullstellensatz for ordinary DAEsystems with constant coefficients,J.Complexity,2014,30(5):588-603.
[63]Gustavson R,Kondratieva M,and Ovchinnikov A,New effective differential Nullstellensatz,Adv.Math.,2016,290:1138-1158.
[64]Ovchinnikov A,Pogudin G,and Scanlon T,Effective difference elimination and Nullstellensatz,ArXiv:1712.01412V2,2017.
[65]Ovchinnikov A,Pogudin G,and Vo T N,Bounds for elimination of unknowns in systems of differential-algebraic equations,ArXiv:1610.0422v6,2018.
[66]Yang L,Zeng Z B,and Zhang W N,Search dependency between algebraic equations:An algorithm applied to automated reasoning,Technical Report ICTP/91/6,International Center For Theoretical Physics,International Atomic Energy Agency,Miramare,Trieste,1991.
[67]Kalkbrener M,A generalized Euclidean algorithm for computing triangular representations of algebraic varieties,J.Symbolic Comput.,1993,15(2):143-167.
[68]Gallo G and Mishra B,Efficient algorithms and bounds for Wu-Ritt characteristic sets,Effective Methods in Algebraic Geometry(Castiglioncello,1990),volume 94 of Progr.Math.,Birkh?user Boston,Boston,MA,1991,119-142.
[69]Wang D M,An elimination method for polynomial systems,J.Symbolic Comput.,1993,16(2):83-114.
[70]Wang D M,Decomposing polynomial systems into simple systems,J.Symbolic Comput.,1998,25(3):295-314.
[71]Wang D M,Computing triangular systems and regular systems,J.Symbolic Comput.,2000,30(2):221-236.
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