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Difference algebra. (English) Zbl 1209.12004
Hazewinkel, M. (ed.), Handbook of algebra. Volume 4. Amsterdam: Elsevier/North-Holland (ISBN 978-0-444-52213-9/hbk). Handbook of Algebra 4, 241-334 (2006).
This is a survey on difference algebra. The author presents basic concepts, difference modules, difference field extensions, difference Galois theory. See also the review of the author’s monograph “Difference algebra” [Algebra and Applications 8. New York, NY: Springer (2008; Zbl 1209.12003)].
For the entire collection see [Zbl 1182.00009].

MSC:
12H10 Difference algebra
12-02 Research exposition (monographs, survey articles) pertaining to field theory
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[1] Aranda-Bricaire, E.; Kotta, U.; Moog, C.H., Linearization of discrete-time systems, SIAM J. control optim., 34, 1999-2023, (1996) · Zbl 0863.93014
[2] Babbitt, A.E., Finitely generated pathological extensions of difference fields, Trans. amer. math. soc., 102, 1, 63-81, (1962) · Zbl 0104.03601
[3] Balaba, I.N., Dimension polynomials of extensions of difference fields, Vestnik moskov. univ., ser. I, mat. mekh., 31-35, (1984), no. 2 (Russian) · Zbl 0543.12017
[4] Balaba, I.N., Calculation of the dimension polynomial of a principal difference ideal, Vestnik moskov. univ., ser. I, mat. mekh., 16-20, (1985), no. 2 (Russian)
[5] Balaba, I.N., Finitely generated extensions of difference fields, (1987), VINITI Moscow, Russia, no. 6632-87 (Russian)
[6] Becker, T.; Weispfenning, V., Gröbner bases. A computational approach to commutative algebra, (1993), Springer-Verlag · Zbl 0772.13010
[7] Bentsen, I., The existence of solutions of abstract partial difference polynomials, Trans. amer. math. soc., 158, 2, 373-397, (1971) · Zbl 0222.12106
[8] Bialynicki-Birula, A., On Galois theory of fields with operators, Amer. J. math., 84, 89-109, (1962) · Zbl 0113.03203
[9] Birkhoff, G.D., General theory of linear difference equations, Trans. amer. math. soc., 12, 243-284, (1911) · JFM 42.0359.02
[10] Birkhoff, G.D., The generalized Riemann problem for linear differential equations and the allied problem for difference and q-difference equations, Proc. natl. acad. sci., 49, 521-568, (1913) · JFM 44.0391.03
[11] Birkhoff, G.D., Note on linear difference and differential equations, Proc. natl. acad. sci., 27, 65-67, (1941) · JFM 67.0305.01
[12] Bronstein, M., On solutions of linear ordinary difference equations in their coefficient field, J. symbolic comput., 29, 6, 841-877, (2000) · Zbl 0961.12004
[13] Buchberger, B., Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal, (1965), University of Innsbruck, Institute for Mathematics, Ph.D. Thesis · Zbl 1245.13020
[14] Casorati, F., Il calcolo delle differenze finite, Ann. mat. pura appl., ser. II, 10, 10-43, (1880-1882) · JFM 12.0305.01
[15] Chatzidakis, Z., A survey on the model theory of difference fields, (), 65-96 · Zbl 0961.03035
[16] Chatzidakis, Z.; Hrushovski, E., Model theory of difference fields, Trans. amer. math. soc., 351, 8, 2997-3071, (1999) · Zbl 0922.03054
[17] Cohn, P.M., Free rings and there relations, (1971), Academic Press · Zbl 0232.16003
[18] Cohn, P.M., Skew fields, Theory of general division rings, (1995), Cambridge Univ. Press · Zbl 0840.16001
[19] Cohn, R.M., Manifolds of difference polynomials, Trans. amer. math. soc., 64, 133-172, (1948) · Zbl 0031.30401
[20] Cohn, R.M., A note on the singular manifolds of a difference polynomial, Bull. amer. math. soc., 54, 917-922, (1948) · Zbl 0037.06602
[21] Cohn, R.M., A theorem on difference polynomials, Bull. amer. math. soc., 55, 595-597, (1949) · Zbl 0033.17902
[22] Cohn, R.M., Inversive difference fields, Bull. amer. math. soc., 55, 597-603, (1949) · Zbl 0033.17903
[23] Cohn, R.M., Singular manifolds of difference polynomials, Ann. math., 53, 445-463, (1951) · Zbl 0042.32202
[24] Cohn, R.M., Extensions of difference fields, Amer. J. math., 74, 507-530, (1952) · Zbl 0046.03801
[25] Cohn, R.M., On extensions of difference fields and the resolvents of prime difference ideals, Proc. amer. math. soc., 3, 178-182, (1952) · Zbl 0046.25802
[26] Cohn, R.M., Essential singular manifolds of difference polynomials, Ann. math., 57, 524-530, (1953) · Zbl 0050.03003
[27] Cohn, R.M., Finitely generated extensions of difference fields, Proc. amer. math. soc., 6, 3-5, (1955) · Zbl 0067.01701
[28] Cohn, R.M., On the intersections of components of a difference polynomial, Proc. amer. math. soc., 6, 42-45, (1955) · Zbl 0067.01702
[29] Cohn, R.M., Specializations over difference fields, Pacific J. math., 5, Suppl. 2, 887-905, (1955) · Zbl 0067.01603
[30] Cohn, R.M., An invariant of difference field extensions, Proc. amer. math. soc., 7, 656-661, (1956) · Zbl 0070.27001
[31] Cohn, R.M., An improved result concerning singular manifolds of difference polynomials, Canad. J. math., 11, 222-234, (1959) · Zbl 0226.12104
[32] Cohn, R.M., Difference algebra, Interscience, (1965) · Zbl 0127.26402
[33] Cohn, R.M., An existence theorem for difference polynomials, Proc. amer. math. soc., 17, 254-261, (1966) · Zbl 0142.28204
[34] Cohn, R.M., Errata to “an existence theorem for difference polynomials”, Proc. amer. math. soc., 18, 1142-1143, (1967) · Zbl 0173.03606
[35] Cohn, R.M., Systems of ideals, Canad. J. math., 21, 783-807, (1969) · Zbl 0179.34203
[36] Cohn, R.M., A difference-differential basis theorem, Canad. J. math., 22, 6, 1224-1237, (1970) · Zbl 0206.05104
[37] Cohn, R.M., Types of singularity of components of difference polynomial, Aequationes math., 9, 2, 236-241, (1973) · Zbl 0265.12103
[38] Cohn, R.M., The general solution of a first order differential polynomial, Proc. amer. math. soc., 55, 1, 14-16, (1976) · Zbl 0327.12103
[39] Cohn, R.M., Solutions in the general solution, (), 117-127, Collection of Papers Dedicated to Ellis Kolchin
[40] Cohn, R.M., Specializations of differential kernels and the Ritt problem, J. algebra, 61, 1, 256-268, (1979) · Zbl 0432.12016
[41] Cohn, R.M., The greenspan bound for the order of differential systems, Proc. amer. math. soc., 79, 4, 523-526, (1980) · Zbl 0436.12018
[42] Cohn, R.M., Order and dimension, Proc. amer. math. soc., 87, 1, 1-6, (1983) · Zbl 0505.12025
[43] Cohn, R.M., Valuations and the Ritt problem, J. algebra, 101, 1, 1-15, (1986) · Zbl 0593.12012
[44] Cohn, R.M., Solutions in the general solution of second order algebraic differential equations, Amer. J. math., 108, 3, 505-523, (1986) · Zbl 0602.12006
[45] Einstein, A., The meaning of relativity, (), 133-165, Appendix II (Generalization of gravitation theory) · Zbl 0063.01229
[46] Eisen, M., Ideal theory and difference algebra, Math. japon., 7, 159-180, (1962) · Zbl 0111.00702
[47] Eisenbud, D., Commutative algebra with view towards algebraic geometry, (1995), Springer-Verlag
[48] Etingof, P.I., Galois groups and connection matrices of q-difference equations, Electron res. announc. amer. math. soc., 1, 1, 1-9, (1995), (electronic)
[49] Evanovich, P., Algebraic extensions of difference fields, Trans. amer. math. soc., 179, 1, 1-22, (1973) · Zbl 0285.12103
[50] Evanovich, P., Finitely generated extensions of partial difference fields, Trans. amer. math. soc., 281, 2, 795-811, (1984) · Zbl 0536.12016
[51] Fliess, M., Esquisses pour une theorie des systemes non lineaires en temps discret, (), 55-67, Special Issue (French)
[52] Fliess, M., Automatique en temps discret et algebre aux differences, Forum math., 2, 3, 213-232, (1990), (French) · Zbl 0706.93039
[53] Fliess, M., A fundamental result on the invertibility of discrete time dynamics, (), 211-223, Lyon, 1990, Progr. Systems Control Theory 8 · Zbl 0789.93095
[54] Fliess, M., Invertibility of causal discrete time dynamical system, J. pure appl. algebra, 86, 2, 173-179, (1993) · Zbl 0780.58040
[55] Fliess, M.; Levine, J.; Martin, P.; Rouchon, O., Differential flatness and defect: an overview, (), 209-225, 1993 · Zbl 1010.93504
[56] Franke, C., Picard-Vessiot theory of linear homogeneous difference equations, Trans. amer. math. soc., 108, 3, 491-515, (1963) · Zbl 0116.02604
[57] Franke, C., Solvability of linear homogeneous difference equations by elementary operations, Proc. amer. math. soc., 17, 240-246, (1964) · Zbl 0143.06102
[58] Franke, C., A note on the Galois theory of linear homogeneous difference equations, Proc. amer. math. soc., 18, 548-551, (1967) · Zbl 0154.03803
[59] Franke, C., The Galois correspondence for linear homogeneous difference equations, Proc. amer. math. soc., 21, 397-401, (1969) · Zbl 0177.30201
[60] Franke, C., Linearly reducible linear difference operators, Aequationes math., 6, 188-194, (1971) · Zbl 0222.12105
[61] Franke, C., Reducible linear difference operators, Aequationes math., 9, 136-144, (1973) · Zbl 0276.12102
[62] Franke, C., A characterization of linear difference equations which are solvable by elementary operations, Aequationes math., 10, 97-104, (1974) · Zbl 0279.12106
[63] Greenspan, B., A bound for the orders of the components of a system of algebraic difference equations, Pacific J. math., 9, 473-486, (1959) · Zbl 0215.36203
[64] Hendrics, P.A., An algorithm for determining the difference Galois group for second order linear difference equations, (1996), Rijksuniversiteit Groningen, Technical Report
[65] Hendrics, P.A., Algebraic aspects of linear differential and difference equations, (1996), Rijksuniversiteit Groningen, Ph.D. Thesis
[66] Hendrics, P.A., An algorithm for computing a standard form for second-order linear q-difference equations, (), 331-352, Eindhoven, 1996, J. Pure Appl. Algebra 117/118
[67] Hendrics, P.A.; Singer, M.F., Solving difference equations in finite terms, J. symbolic comput., 27, 239-259, (1999) · Zbl 0930.39004
[68] Herzog, F., Systems of algebraic mixed difference equations, Trans. amer. math. soc., 37, 286-300, (1935) · JFM 61.0501.02
[69] Hrushovski, E., The Manin-mamford conjecture and the model theory of difference fields, Ann. pure appl. logic, 112, 1, 43-115, (2001) · Zbl 0987.03036
[70] Infante, R.P., Strong normality and normality for difference fields, Aequationes math., 20, 121-122, (1980) · Zbl 0459.12018
[71] Infante, R.P., Strong normality and normality for difference fields, Aequationes math., 20, 159-165, (1980) · Zbl 0459.12018
[72] Infante, R.P., The structure of strongly normal difference extensions, Aequationes math., 21, 1, 16-19, (1980) · Zbl 0459.12019
[73] Infante, R.P., On the Galois theory of difference fields, Aequationes math., 22, 112-113, (1981)
[74] Infante, R.P., On the Galois theory of difference fields, Aequationes math., 22, 194-207, (1981) · Zbl 0483.12005
[75] Infante, R.P., On the inverse problem in difference Galois theory, Contemp. math., 13, 349-352, (1982), Algebraists’ Homage: Papers in the Ring Theory and Related Topics, New Haven, Conn., 1981 · Zbl 0509.12024
[76] Iwasawa, K.; Tamagawa, T., On the group of automorphisms of a functional field, J. math. soc. Japan, 3, 137-147, (1951) · Zbl 0044.26901
[77] Jacobi, C.G.J., Gesammelte werke, (), 191-216, Vol. 5
[78] Johnson, J.L., Differential dimension polynomials and a fundamental theorem on differential modules, Amer. J. math., 91, 1, 239-248, (1969) · Zbl 0179.34303
[79] Johnson, J.L., Kähler differentials and differential algebra, Ann. of math., 89, 2, 92-98, (1969) · Zbl 0179.34302
[80] Johnson, J.L., A note on Krull dimension for differential rings, Comment. math. helv., 44, 207-216, (1969) · Zbl 0179.34401
[81] Johnson, J.L., Kähler differentials and differential algebra in arbitrary characteristic, Trans. amer. math. soc., 192, 201-208, (1974) · Zbl 0288.12104
[82] Johnson, J.L.; Sit, W., On the differential transcendence polynomials of finitely generated differential field extensions, Amer. J. math., 101, 1249-1263, (1979) · Zbl 0434.12011
[83] Karr, M., , summation in finite terms, J. assoc. comput. Mach., 28, 2, 305-350, (1981) · Zbl 0494.68044
[84] Kolchin, E.R., The notion of dimension in the theory of algebraic differential equations, Bull. amer. math. soc., 70, 570-573, (1964) · Zbl 0144.03702
[85] Kolchin, E.R., Some problems in differential algebra, (), 269-276, 1966
[86] Kolchin, E.R., Differential algebra and algebraic groups, (1973), Academic Press · Zbl 0264.12102
[87] Kondrateva, M.V.; Levin, A.B.; Mikhalev, A.V.; Pankratev, E.V., Computation of dimension polynomials, Internat. J. algebra comput., 2, 2, 117-137, (1992) · Zbl 0774.12007
[88] Kondrateva, M.V.; Levin, A.B.; Mikhalev, A.V.; Pankratev, E.V., Differential and difference dimension polynomials, (1998), Kluwer Acad. Publ
[89] Kondrateva, M.V.; Pankratev, E.V., A recursive algorithm for the computation of Hilbert polynomial, (), 365-375, Lecture Notes in Comput. Sci. 378
[90] Kondrateva, M.V.; Pankratev, E.V., Algorithms of computation of characteristic Hilbert polynomials, (), 129-146, (in Russian)
[91] Kondrateva, M.V.; Pankratev, E.V.; Serov, R.E., Computations in differential and difference modules, (), 208-213, (in Russian)
[92] Kowalski, P.; Pillay, A., A note on groups definable in difference fields, Proc. amer. math. soc., 130, 1, 205-212, (2002), (electronic) · Zbl 0983.03033
[93] Kreimer, H.F., The foundations for extension of differential algebra, Trans. amer. math. soc., 111, 482-492, (1964) · Zbl 0142.01104
[94] Kreimer, H.F., An extension of differential Galois theory, Trans. amer. math. soc., 118, 247-256, (1965) · Zbl 0144.03604
[95] Kreimer, H.F., On an extension of the Picard-Vessiot theory, Pacific J. math., 15, 191-205, (1965) · Zbl 0143.06103
[96] Lando, B., Jacobi’s bound for first order difference equations, Proc. amer. math. soc., 32, 1, 8-12, (1972) · Zbl 0241.12108
[97] Lando, B., Extensions of difference specializations, Proc. amer. math. soc., 79, 2, 197-202, (1980) · Zbl 0446.12019
[98] Latyshev, V.M.; Mikhalev, A.V.; Pankratev, E.V., Construction of canonical simplifiers in modules over rings of polynomials, Visnik kiev univ., ser. mat. mekh., 27, 65-67, (1985), (Ukrainian) · Zbl 0900.13001
[99] Levin, A.B., Characteristic polynomials of filtered difference modules and difference field extensions, Uspekhi mat. nauk, 33, 3, 177-178, (1978), (Russian). English transl.: Russian Math. Surveys 33 (3) (1978), 165-166 · Zbl 0413.12024
[100] Levin, A.B., Characteristic polynomials of inversive difference modules and some properties of inversive difference dimension, Uspekhi mat. nauk, 35, 1, 201-202, (1980), (Russian). English transl.: Russian Math. Surveys 35 (1) (1980), 217-218 · Zbl 0426.12017
[101] Levin, A.B., Characteristic polynomials of difference modules and some properties of difference dimension, (1980), VINITI Moscow, no. 2175-80 (Russian)
[102] Levin, A.B., Type and dimension of inversive difference vector spaces and difference algebras, (1982), VINITI Moscow, no. 1606-82 (Russian)
[103] Levin, A.B., Characteristic polynomials of δ-modules and finitely generated δ-field extensions, (1985), VINITI Moscow, no. 334-85 (Russian)
[104] Levin, A.B., Inversive difference modules and problems of solvability of systems of linear difference equations, (1985), VINITI Moscow, no. 335-85 (Russian)
[105] Levin, A.B., Computation of Hilbert polynomials in two variables, J. symbolic comput., 28, 681-709, (1999) · Zbl 0976.13011
[106] Levin, A.B., Characteristic polynomials of finitely generated modules over Weyl algebras, Bull. austral. math. soc., 61, 387-403, (2000) · Zbl 0968.16012
[107] Levin, A.B., Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials, J. symbolic comput., 29, 1-26, (2000)
[108] Levin, A.B., On the set of Hilbert polynomials, Bull. austral. math. soc., 64, 291-305, (2001) · Zbl 1096.13520
[109] A.B. Levin, Multivariable dimension polynomials and new invariants of differential field extensions. Internat. J. Math. Math. Sci. {\bf27} (4), 201-213. · Zbl 1010.12006
[110] Levin, A.B.; Mikhalev, A.V., Difference-differential dimension polynomials, (1988), VINITI Moscow, no. 6848-B88 (Russian)
[111] Levin, A.B.; Mikhalev, A.V., Dimension polynomials of filtered G-modules and finitely generated G-field extensions, (), 74-94 · Zbl 0745.12007
[112] Levin, A.B.; Mikhalev, A.V., Type and dimension of finitely generated vector G-spaces, Vestnik moskov. univ., ser. I, mat. mekh., 72-74, (1991), no. 4 (Russian). English transl.: Moscow Univ. Math. Bull. 46 (4), 51-52 · Zbl 0745.12007
[113] Levin, A.B.; Mikhalev, A.V., Dimension polynomials of difference-differential modules and of difference-differential field extensions, Abelian groups and modules, 10, 56-82, (1991), (Russian)
[114] Levin, A.B.; Mikhalev, A.V., Dimension polynomials of filtered differential G-modules and extensions of differential G-fields, Contemp. math., 131, 2, 469-489, (1992) · Zbl 0759.12003
[115] Levin, A.B.; Mikhalev, A.V., Type and dimension of finitely generated G-algebras, Contemp. math., 184, 275-280, (1995) · Zbl 0840.13004
[116] Macintyre, A., Generic automorphisms of fields, Ann. pure appl. logic, 88, 2-3, 165-180, (1997) · Zbl 0891.03015
[117] Mikhalev, A.V.; Pankratev, E.V., Differential dimension polynomial of a system of differential equations, (), 57-67, (Russian).
[118] Mikhalev, A.V.; Pankratev, E.V., Differential and difference algebra, (), 67-139, Itogi Nauki i Tekhniki, Akad. Nauk SSSR (Russian). English transl.: J. Soviet Math. 45 (1) (1989), 912-955
[119] Mikhalev, A.V.; Pankratev, E.V., Computer algebra. calculations in differential and difference algebra, (1989), Moscow State Univ., (Russian)
[120] Mishra, B., Algorithmic algebra, (1993), Springer-Verlag · Zbl 0804.13009
[121] Moosa, R., On difference fields with quantifier elimination, Bull. London math. soc., 33, 6, 641-646, (2001) · Zbl 1022.03019
[122] Nishioka, K., A note on differentially algebraic solutions of first order linear difference equations, Aequationes math., 27, 32-48, (1984) · Zbl 0542.12012
[123] Nonvide, S., Corps aux différences finies, C. R. acad. sci. Paris Sér. I math., 314, 6, 423-425, (1992), (French) · Zbl 0757.12005
[124] Pankratev, E.V., The inverse Galois problem for the extensions of difference fields, Algebra i logika, 11, 87-118, (1972), (Russian). English transl.: Algebra and Logic 11 (1972), 51-69
[125] Pankratev, E.V., The inverse Galois problem for extensions of difference fields, Uspekhi mat. nauk, 27, 1, 249-250, (1972), (Russian) · Zbl 0229.12103
[126] Pankratev, E.V., Fuchsian difference modules, Uspekhi mat. nauk, 28, 3, 193-194, (1973), (Russian)
[127] Pankratev, E.V., Computations in differential and difference modules, symmetries of partial differential equations, part III, Acta appl. math., 16, 2, 167-189, (1989)
[128] Petkovsek, M., Finding closed form solutions of difference equations by symbolic methods, (1990), Carnegie Mellon University, Thesis, Dept. of Comp. Sci.
[129] Pillay, A., A note on existentially closed difference fields with algebraically closed fixed field, J. symbolic logic, 66, 2, 719-721, (2001) · Zbl 0991.03038
[130] Praagman, C., The formal classification of linear difference operators, Proc. kon. ned. ac. wet., ser. A, 86, 249-261, (1983) · Zbl 0519.39003
[131] Praagman, C., Meromorphic linear difference equations, (1985), University of Groningen, Thesis · Zbl 0586.30024
[132] Riordan, J., Combinatorial identities, (1968), Wiley · Zbl 0194.00502
[133] Ritt, J.F., Algebraic difference equations, Bull. amer. math. soc., 40, 303-308, (1934) · JFM 60.0401.03
[134] Ritt, J.F., Complete difference ideals, Amer. J. math., 63, 681-690, (1941) · Zbl 0026.05902
[135] Ritt, J.F., Differential algebra, Amer. math. soc. coll. publ. 33, (1973), Amer. Math. Soc.
[136] Ritt, J.F.; Doob, J.L., Systems of algebraic difference equations, Amer. J. math., 55, 505-514, (1933) · JFM 59.0456.01
[137] Ritt, J.F.; Raudenbush, H.W., Ideal theory and algebraic difference equations, Trans. amer. math. soc., 46, 445-453, (1939) · Zbl 0022.10603
[138] Scanlon, T.; Voloch, J.F., Difference algebraic subgroups of commutative algebraic groups over finite fields, Manuscripta math., 99, 3, 329-339, (1999) · Zbl 0974.12013
[139] Sit, W., Well-ordering of certain numerical polynomials, Trans. amer. math. soc., 212, 37-45, (1975) · Zbl 0295.12103
[140] Strodt, W., Systems of algebraic partial difference equations, Master essay, (1937), Columbia Univ.
[141] Strodt, W., Principal solutions of difference equations, Amer. J. math., 69, 717-757, (1947) · Zbl 0034.19403
[142] van der Put, M.; Singer, M.F., Galois theory of difference equations, (1997), Springer · Zbl 0930.12006
[143] van Hoeij, M., Rational solutions of linear difference equations, (1998), Dept. of Mathematics, Florida State University, Technical Report · Zbl 0919.65088
[144] Zariski, O.; Samuel, P., Commutative algebra, (1958), Van Nostrand, Vol. I Vol. II (1960) · Zbl 0121.27901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.