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GCD Domains, Gauss’ Lemma, and Contents of Polynomials

  • D. D. Anderson
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

The purpose of this article is to survey the work done on GCD domains and their generalizations. While the best known examples of GCD domains are UFD’s and Bezout domains, we concentrate on GCD domains that are not UFD’s or Bezout domains as there is already an extensive literature on UFD’s and Bezout domains including survey articles [44], [100] and books [98] and [53]. Among the generalizations of GCD domains surveyed are Schreier domains, Prüfer υ-multiplication domains (PVMD’s) and general­ized GCD domains (G-GCD domains).

Keywords

Prime Ideal Integral Domain Finite Type Valuation Domain Quotient Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    H.E. Adams, Factorization-prime ideals in integral domains, Pacific Math. 66 (1976), 1–8.MATHCrossRefGoogle Scholar
  2. [2]
    D.D Anderson, Another generalization of principal ideal rings, J. Algebra 48 (1977), 409–416.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    D.D. Anderson, On a result of M. Martin concerning the number of generators of products of invertible ideals, Comm. Algebra 15 (1988), 1765–1768.CrossRefGoogle Scholar
  4. [4]
    D.D. Anderson, Star-operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    D.D. Anderson, Extensions of unique factorization: A survey, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 205 (1999), 31–53.Google Scholar
  6. [6]
    D.D. Anderson and D.F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1979), 215–221.Google Scholar
  7. [7]
    D.D. Anderson and D.F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), 196–215.MATHGoogle Scholar
  8. [8]
    D.D. Anderson and D.F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), 549–569.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    D.D. Anderson and D.F. Anderson, The ring R[X,r/X], Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 171 (1995), 95–113.Google Scholar
  10. [10]
    D.D. Anderson, D.F. Anderson, and J. Park, GCD sets in integral domains, Houston J. Math. 25 (1999), 15–34.MathSciNetMATHGoogle Scholar
  11. [11]
    D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    D.D. Anderson, D.F. Anderson, and M. Zafrullah, A generalization of unique factor­ization, Bollettino U.M.I. (7) 9-A(1995), 401–413.MathSciNetGoogle Scholar
  14. [14]
    D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    D.D. Anderson and B.G. Kang, Pseudo-Dedekind domains and divisorial ideals in R[X] T, J. Algebra 122 (1989), 323–336.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    D.D. Anderson and B.G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), 82–94.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    D.D. Anderson and B.G. Kang, Formally integrally closed domains and the rings R((X)) and R{(X)}, J. Algebra 200 (1998), 347–362.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    D.D. Anderson, K.R. Knopp, and R.L. Lewin, Almost Bezout domains, II, J. Algebra 167 (1994), 547–556.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    D.D. Anderson, D.J. Kwak, and M. Zafrullah, Agreeable domains, Comm. Algebra 23 (1995), 4861–4883.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    D.D. Anderson and L.A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141–154.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    D.D. Anderson and R. Markanda, Unique factorization rings with zero divisors, Hous­ton J. Math. 11 (1985), 15–30 and Corrigendum, 423–426.MathSciNetMATHGoogle Scholar
  22. [22]
    D.D. Anderson, J.L. Mott, and M. Zafrullah, Some quotient based statements in mul­tiplicative ideal theory, Bollettino U.M.I. (7) 3-B (1989), 455–476.MathSciNetGoogle Scholar
  23. [23]
    D.D. Anderson and R.O. Quintero, Some generalizations of GCD domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 189 (1997), 189–195.MathSciNetGoogle Scholar
  24. [24]
    D.D. Anderson and S. Valdes-Leon, Factorization in commutative rings with zero divisors, Rocky Mountain J. Math. 26 (1996), 439–480.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    D.D. Anderson and M. Zafrullah, Almost Bezout rings, J. Algebra 142 (1991), 285–309.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    D.D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Bollettino U.M.I. (7) 8-A (1994), 397–402.MathSciNetGoogle Scholar
  27. [27]
    D.D. Anderson and M. Zafrullah, P.M. Cohn’s completely primal elements, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 171 (1995), 115–123.MathSciNetGoogle Scholar
  28. [28]
    D.D. Anderson and M. Zafrullah, Star operations and primitive polynomials, Comm. Algebra 27 (1999), 3137–3142.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    D.F. Anderson, Integral v-ideals, Glasgow Math. J. 22 (1981), 167–172.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    D.F. Anderson and J. Ohm, Valuations and semi-valuations of graded domains, Math. Ann. 256 (1981), 145–156.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    M. Anderson and J. Watkins, Coherence of power series over pseudo-Bezout domains, J. Algebra 107 (1987), 187–194.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    J.T. Arnold and J.W. Brewer, Kronecker function rings and flat D[X]-modules, Proc. Amer. Math. Soc. 27 (1971), 326–330.MathSciNetGoogle Scholar
  33. [33]
    J.T. Arnold and R. Gilmer, On the contents of polynomials, Proc. Amer. Math. Soc. 24 (1970), 556–562.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    J.T. Arnold and P.B. Sheldon, Integral domains that satisfy Gauss’s Lemma, Michigan Math. J. 22 (1975), 39–51.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    A. Badawi, On domains which have prime ideals that are linearly ordered, Comm. Algebra 23 (1995), 4365–4373.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20 (1973), 79–95.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    D. Boccioni, Alcune osservazioni sugli anelli pseudo-bezoutiani e fattoriali, Rend. Sem. Mat. Univ. Padova 37 (1967), 273–288.MathSciNetMATHGoogle Scholar
  38. [38]
    N. Bourbaki, Commutative Algebra, Addison-Wesley Publishing Company, Reading, MA, 1972.MATHGoogle Scholar
  39. [39]
    A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Grèce 29 (1988), 45–59.MathSciNetMATHGoogle Scholar
  40. [40]
    J.W. Brewer and D.L. Costa, Projective modules over some non-Noetherian polyno­mial rings, J. Pure Appl. Algebra 13 (1978), 157–163.MathSciNetCrossRefGoogle Scholar
  41. [41]
    J.W. Brewer and E.A. Rutter, D + M constructions with general overrings, Michigan Math. J. 23 (1976), 33–42.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    W. Bruns and A. Guerrieri, The Dedekind-Mertens Formula and determinantal ideals, preprint.Google Scholar
  43. [43]
    P.M. Cohn, Bezout rings and their subrings, Math. Proc. Cambridge Philos. Soc. 64 (1968), 251–264.MATHCrossRefGoogle Scholar
  44. [44]
    P.M. Cohn, Unique factorization domains, Amer. Math. Monthly 80 (1973), 1–18.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    A. Corso, W. Heinzer, and C. Huneke, A generalized Dedekind-Mertens lemma and its converse, Trans. Amer. Math. Soc. 350 (1998), 5095–5109.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    A. Corso, W. Vasconcelos, and R.H. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra 125 (1998), 117–127.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    D. Costa, J.L. Mott, and M. Zafrullah, The construction D + XD S [X], J. Algebra 53 (1978), 423–439.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    J. Dawson and D.E. Dobbs, On going down in polynomial rings, Canad. J. Math. 26 (1974), 177–184.MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    D.M. Dribin, Prüfer ideals in commutative rings, Duke Math. J. 4 (1938), 737–751.MathSciNetCrossRefGoogle Scholar
  50. [50]
    H.M. Edwards, Divisor Theory, Birkäuser, Boston, Basel and Berlin, 1990.MATHGoogle Scholar
  51. [51]
    H. Flanders, A remark on Kronecker’s Theorem on forms, Proc. Amer. Math. Soc. 3 (1952), 197.MathSciNetMATHGoogle Scholar
  52. [52]
    H. Flanders, The meaning of the form calculus in classical ideal theory, Trans. Amer. Math. Soc. 95 (1960), 92–100.MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    M. Fontana, J.A. Huckaba, and I.J. Papick, Prüfer Domains, Marcel Dekker, New York, 1997.MATHGoogle Scholar
  54. [54]
    L. Fuchs, Riesz groups, Ann. Scuola Norm. Sup. Pisa 19 (1965), 1–34.MathSciNetMATHGoogle Scholar
  55. [55]
    C.F. Gauss, Disquisitiones Arithmeticae, Springer-Verlag, New York, 1986. Translated by Arthur Clarke.MATHGoogle Scholar
  56. [56]
    R. Gilmer, Some applications of the Hilfsatz von Dedekind-Mertens, Math. Scand. 20 (1967), 240–244.MathSciNetMATHGoogle Scholar
  57. [57]
    R. Gilmer, An embedding theorem for HCF-rings, Math. Proc. Cambridge Philos. Soc. 68 (1970), 583–587.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathe­matics, vol. 91, Queen’s University, Kingston, Ontario, 1992.MATHGoogle Scholar
  59. [59]
    R. Gilmer, A. Grams, and T. Parker, Zero divisors in power series rings, J. Reine Angew. Math. 278/279 (1975), 145–164.MathSciNetGoogle Scholar
  60. [60]
    R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    S. Glaz, Finite conductor rings, Proc. Amer. Math. Soc, to appear.Google Scholar
  62. [62]
    S. Glaz and W. Vasconcelos, Gaussian polynomials, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 186 (1997), 325–337.MathSciNetGoogle Scholar
  63. [63]
    S. Glaz and W. Vasconcelos, The content of Gaussian polynomials, J. Algebra 202 (1998), 1–9.MathSciNetMATHCrossRefGoogle Scholar
  64. [64]
    K.R. Goodearl, Partially Ordered Abelian Groups With Interpolation, Mathematical Surveys and Monographs 20, American Mathematical Society, Providence, RI, 1986.MATHGoogle Scholar
  65. [65]
    M. Griffin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710–722.MathSciNetMATHCrossRefGoogle Scholar
  66. [66]
    F. Halter-Koch, Ideal Systems: An Introduction to Multiplicative Ideal Theory, Marcel Dekker, New York, 1998.MATHGoogle Scholar
  67. [67]
    W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739–745.MathSciNetMATHCrossRefGoogle Scholar
  68. [68]
    W. Heinzer and C. Huneke, The Dedekind-Mertens Lemma and the contents of poly­nomials, Proc. Amer. Math. Soc. 126 (1998), 1305–1309.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    P. Jaffard, Contribution à la théorie des groupes ordonnés, J. Math. Pures Appl. 32 (1953), 203–280.MathSciNetMATHGoogle Scholar
  70. [70]
    P. Jaffard, Les Systèmes d’Idéaux, Dunod, Paris, 1960.MATHGoogle Scholar
  71. [71]
    B.G. Kang, Prüfer v-multiplication domains and the ring R[X]jy v, J. Algebra 123 (1989), 151–170.MathSciNetMATHCrossRefGoogle Scholar
  72. [72]
    I. Kaplansky, Commutative Rings (revised edition), University of Chicago Press, Chicago, 1974.MATHGoogle Scholar
  73. [73]
    W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche, Math. Z. 41 (1936), 545–577.MathSciNetCrossRefGoogle Scholar
  74. [74]
    W. Krull, Idealtheorie, Chelsea, New York, 1948.Google Scholar
  75. [75]
    L.R. LeRiche, The ring R(X), J. Algebra 67 (1980), 327–341.MathSciNetCrossRefGoogle Scholar
  76. [76]
    R.L. Lewin, Almost generalized GCD-domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 189 (1997), 371–382.MathSciNetGoogle Scholar
  77. [77]
    A. Loper, Two Prüfer domain counterexamples, J. Algebra 221(1999), 630–643.MathSciNetMATHCrossRefGoogle Scholar
  78. [78]
    F. Lucuis, Rings with a theory of greatest common divisors, Manuscripta Math. 95 (1998), 117–136.MathSciNetCrossRefGoogle Scholar
  79. [79]
    S. Malik, J.L Mott, and M. Zafrullah, The G CD property and irreducible quadratic polynomials, Internat. J. Math. & Math. Sei. 9 (1986), 749–752.MathSciNetMATHCrossRefGoogle Scholar
  80. [80]
    R. Matsuda, On the content condition of a graded integral domain, Comment. Math. Univ. St. Pauli 33 (1984), 79–86.MathSciNetMATHGoogle Scholar
  81. [81]
    R. Matsuda, Note on integral domains that satisfy Gauss’s Lemma, Math. Japonica 41 (1995), 625–630.MathSciNetMATHGoogle Scholar
  82. [82]
    R. Matsuda, Note on Schreier semigroup rings, Math. J. Okayama Univ. 39 (1997), 41–44.MathSciNetMATHGoogle Scholar
  83. [83]
    S. McAdam, Two conductor theorems, J. Algebra 23 (1972), 239–240.MathSciNetMATHCrossRefGoogle Scholar
  84. [84]
    S. McAdam and D.E. Rush, Schreier rings, Bull. London Math. Soc. 10 (1978), 77–80.MathSciNetMATHCrossRefGoogle Scholar
  85. [85]
    J.L. Mott, The group of divisibility and its applications, Conference on Commutative Algebra, Kansas 1972, Lecture Notes in Mathematics, No. 311, Springer-Verlag, New York, 1973.Google Scholar
  86. [86]
    J.L. Mott, Convex directed subgroups of a group of divisibility, Canad. J. Math. 26 (1974), 532–542.MathSciNetMATHGoogle Scholar
  87. [87]
    J.L. Mott, B. Nashier, and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra 18 (1990), 1569–1583.MathSciNetMATHCrossRefGoogle Scholar
  88. [88]
    J.L. Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388–401.MathSciNetGoogle Scholar
  89. [89]
    J.L. Mott and M. Zafrullah, On Prüfer v-multiplication domains, Manuscripts Math. 35 (1981), 1–26.MathSciNetMATHCrossRefGoogle Scholar
  90. [90]
    D.G. Northcott, A generalization of a theorem on the content of polynomials, Math. Proc. Cambridge Philos. Soc. 55 (1959), 282–288.MathSciNetMATHCrossRefGoogle Scholar
  91. [91]
    J. Pahikkala, Some formulae for multiplying and inverting ideals, Ann. Univ. Turku. Ser AI No. 183 (1982), 11 pp.MathSciNetGoogle Scholar
  92. [92]
    G. Picavet, About G CD domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, vol. 205 (1999), 501–519.MathSciNetGoogle Scholar
  93. [93]
    H. Prüfer, Untersuchungen über Teilbarkeitseigenschaften in Körpern, J. Reine Angew. Math. 168 (1932), 1–36.Google Scholar
  94. [94]
    J. Querre, Idéaux divisoriels d’un anneau de polynômes, J. Algebra 64 (1980), 270–284.MathSciNetMATHCrossRefGoogle Scholar
  95. [95]
    M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sei. 73 (1997), 14–17.MathSciNetMATHCrossRefGoogle Scholar
  96. [96]
    M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187–199.MathSciNetMATHCrossRefGoogle Scholar
  97. [97]
    D. Rush, Content algebras, Canad. Math. Bull. 21 (1978), 329–334.MathSciNetMATHCrossRefGoogle Scholar
  98. [98]
    P. Samuel, Unique Factorization Domains, Tata Institute of Fundamental Research, Bombay, 1964.Google Scholar
  99. [99]
    P. Samuel, On a construction of P.M. Cohn, Math. Proc. Cambridge Philos. Soc. 64 (1968), 249–250.MathSciNetMATHCrossRefGoogle Scholar
  100. [100]
    P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945–952.MathSciNetMATHCrossRefGoogle Scholar
  101. [101]
    P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282–301.MathSciNetMATHCrossRefGoogle Scholar
  102. [102]
    J. Shamash and S.T. Smith, Embedding GCD domains in Bézout domains, J. London Math. Soc. 48 (1993), 15–30, Addendum: 54 (1996), 209.MathSciNetMATHCrossRefGoogle Scholar
  103. [103]
    P.B. Sheldon, Two counterexamples involving complete integral closure in finite-dimensional Prüfer domains, J. Algebra 27 (1973), 462–474.MathSciNetMATHCrossRefGoogle Scholar
  104. [104]
    P.B. Sheldon, Prime ideals in GCD-domains, Canad. J. Math. 6 (1974), 98–107.MathSciNetCrossRefGoogle Scholar
  105. [105]
    T. Skolem, Eine Bemerkung über gewisse Ringe mit Anwendung auf die Produktzer­legung von Polynomen, Norsk Mat. Tidsskr. 21 (1939), 99–107.MathSciNetGoogle Scholar
  106. [106]
    S.T. Smith, Fermat’s last theorem and Bézout’s Theorem in GCD domains, J. Pure Appl. Algebra 79 (1992), 63–85.MathSciNetMATHCrossRefGoogle Scholar
  107. [107]
    S.T. Smith, Building discretely ordered Bézout domains and GCD domains, J. Algebra 159 (1993), 191–239.MathSciNetMATHCrossRefGoogle Scholar
  108. [108]
    U. Storch, Fastfaktorielle Ringe, Sehr. Math. Inst. Univ. Münster, 36 (1967).MATHGoogle Scholar
  109. [109]
    H.T. Tang, Gauss’s Lemma, Proc. Amer. Math. Soc. 35 (1972), 372–376.MathSciNetMATHGoogle Scholar
  110. [110]
    H. Tsang, Gauss’s Lemma, Dissertation, University of Chicago, Chicago, 1965.Google Scholar
  111. [Ill]
    A.I. Uzkov, Additional information concerning the content of the product of polynomi­als, Math. Notes 16 (1974), 825–827 (English translation of Math. Zametki 16 (1974), 395–398).Google Scholar
  112. [112]
    W. Vasconcelos, The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381–386.MathSciNetMATHCrossRefGoogle Scholar
  113. [113]
    M. Zafrullah, Semirigid GCD domains, Manuscripta Math. 17 (1975), 55–66.MathSciNetMATHCrossRefGoogle Scholar
  114. [114]
    M. Zafrullah, Rigid elements in GCD domains, J. Natur. Sei. and Math. 17 (1977), 7–14.MathSciNetMATHGoogle Scholar
  115. [115]
    M. Zafrullah, Unique representation domains, J. Natur. Sei. and Math. 18 (1978), 19–29.MathSciNetMATHGoogle Scholar
  116. [116]
    M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), 191–204.MathSciNetMATHCrossRefGoogle Scholar
  117. [117]
    M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62.MathSciNetMATHCrossRefGoogle Scholar
  118. [118]
    M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920.MathSciNetMATHCrossRefGoogle Scholar
  119. [119]
    M. Zafrullah, Some polynomial characterizations of Prüfer v-multiplication domains, J. Pure Appl. Algebra 32 (1988), 231–237.MathSciNetCrossRefGoogle Scholar
  120. [120]
    M. Zafrullah, The D + XDs[X] construction from GCD-domains, J. Pure Appl. Al­gebra 50 (1988), 93–107.MathSciNetMATHCrossRefGoogle Scholar
  121. [121]
    M. Zafrullah, On Riesz groups, Manuscripta Math. 80 (1993), 225–238.MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • D. D. Anderson
    • 1
  1. 1.Department of MathematicsThe University of IowaIowa CityUSA

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