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Title

On simultaneous diagonal equations and inequalities

Authors 1, 2

Affiliations

  1. Mathematisches Institut, Bunsenstrasse 3-5, 3400 Göttingen, Germany
  2. Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield S3 7RH, England

Bibliography

  1. M. Aigner, Combinatorial Theory, Springer, Berlin 1979.
  2. O. D. Atkinson, J. Brüdern and R. J. Cook, Three additive cubic equations, Acta Arith. 60 (1991), 29-83.
  3. O. D. Atkinson, J. Brüdern and R. J. Cook, Simultaneous additive congruences to a large prime modulus, Mathematika, to appear.
  4. R. C. Baker, Diagonal cubic equations, I , in: Théorie des Nombres, Number Theory, J.-M. De Koninck and C. Levesque (eds.), de Gruyter, Berlin 1989, 15-28.
  5. R. C. Baker, Diagonal cubic equations, II , Acta Arith. 53 (1989), 217-250.
  6. R. C. Baker, Diagonal cubic equations, III , Proc. London Math. Soc. (3) 58 (1989), 495-518.
  7. R. C. Baker and J. Brüdern, On pairs of additive cubic equations, J. Reine Angew. Math. 391 (1988), 157-180.
  8. B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102-105.
  9. J. Brüdern, Additive diophantine inequalities with mixed powers I, Mathematika 34 (1987), 124-130.
  10. J. Brüdern, Cubic diophantine inequalities, Mathematika 35 (1988), 51-58.
  11. J. Brüdern, On pairs of diagonal cubic forms, Proc. London Math. Soc. (3) 61 (1990), 273-343.
  12. J. Brüdern, On Waring's problem for fifth powers and some related topics, Proc. London Math. Soc. 457-479.
  13. J. Brüdernand R. J. Cook, On pairs of cubic diophantine inequalities, Mathematika 38 (1991), 250-263.
  14. J. Brüdernand R. J. Cook, Systems of cubic equations, in preparation.
  15. J. W. S. Cassels and M. J. T. Guy, On the Hasse principle for cubic surfaces, Mathematika 13 (1966), 111-120.
  16. R. J. Cook, Simultaneous quadratic equations, J. London Math. Soc. (2) 4 (1971), 319-326.
  17. R. J. Cook, A note on a lemma of Hua, Quart. J. Math. Oxford 23 (1972), 287-288.
  18. R. J. Cook, Pairs of additive equations, Michigan Math. J. 19 (1972), 325-331.
  19. R. J. Cook, Simultaneous quadratic equations II, Acta Arith. 25 (1973), 1-5.
  20. R. J. Cook, Simultaneous quadratic inequalities, Acta Arith. 25 (1974), 337-346.
  21. R. J. Cook, Pairs of additive equations II. Large odd degree, J. Number Theory 17 (1983), 80-92.
  22. R. J. Cook, Pairs of additive equations III: quintic equations, Proc. Edinburgh Math. Soc. (2) 26 (1983), 191-211.
  23. R. J. Cook, Pairs of additive equations IV. Sextic equations, Acta Arith. 43 (1984), 227-243.
  24. R. J. Cook, Pairs of additive congruences: quintic congruences II, in: Computers in Mathematical Research, N. M. Stephens and M. P. Thorne (eds.), Clarendon Press, Oxford 1988, 93-117.
  25. H. Davenport, Indefinite quadratic forms in many variables, Mathematika 3 (1956), 81-101.
  26. H. Davenport, Cubic forms in thirty-two variables, Philos. Trans. Roy. Soc. London Ser. A 251 (1959), 193-232.
  27. H. Davenport and D. J. Lewis, Homogeneous additive equations, Proc. Roy. Soc. London Ser. A 274 (1963), 443-460.
  28. H. Davenport and D. J. Lewis, Cubic equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 261 (1966), 97-136.
  29. H. Davenport and D. J. Lewis, Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557-595.
  30. H. Davenport and K. F. Roth, The solubility of certain Diophantine inequalities, Mathematika 2 (1955), 81-96.
  31. E. Dörner, Simultaneous diagonal equations over certain p-adic fields, J. Number Theory 36 (1990), 1-11.
  32. L. Low, J. Pitman and A. Wolff, Simultaneous diagonal congruences, J. Number Theory 29 (1988), 31-59.
  33. T. Nadesalingam, Diophantine inequalities in many variables, Ph.D. thesis, Univ. of Adelaide, South Australia, 1981.
  34. J. Pitman, Pairs of diagonal inequalities, in: Recent Progress in Analytic Number Theory, Vol. 2, H. Halberstam and C. Hooley (eds.), Academic Press, London 1981, 183-215.
  35. W. M. Schmidt, Linear forms with algebraic coefficients. I, J. Number Theory 3 (1971), 253-277.
  36. W. M. Schmidt, Diophantine inequalities for forms of odd degree, Adv. in Math. 38 (1980), 128-151.
  37. W. M. Schmidt, The solubility of certain p-adic equations, J. Number Theory 19 (1984), 63-80.
  38. R. C. Vaughan, On pairs of additive cubic equations, Proc. London Math. Soc. (3) 34 (1977), 354-364.
  39. R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press, 1981.
  40. R. C. Vaughan, A new iterative method in Waring's problem, Acta Math. 162 (1989), 1-71.
  41. R. C. Vaughan and T. Wooley, On Waring's problem: some refinements, Proc. London Math. Soc. (3) 63 (1991), 35-68.
Acta Arithmetica
1992/62/2
Export to PBN, Opens in new tab
Pages:
125-149
Main language of publication
English
Received
1991-10-04
Published
1992
Exact and natural sciences