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Appendix: Open Problems Appendix: Open Problems Some currently unsolved problems were marked with asterisks at the end of several of the preceding chapters. Most of them seem to be at least approach­ able. No such problems are listed at the end of Chapter 14. Indeed, for the problems left open in Chapter 14 it is hardly possible at the present time to distinguish among those that may be solved with moderate effort and those that are completely hopeless. Instead, it appears preferable to list here a few such problems whose solution appears important, without any suggestion concerning the depth of the problem, or its degree of difficulty. 1. Let us formulate the Hasse principle as follows: "For a certain class ~ of polynomial equations in n variables over the rational field, solvability in 0 is equivalent to solvability in all 0P' for p ,,:; 00." The problem is to define as accurately as possible the extent of the class ~. 2. As seen, Artin and Pfister have solved Hilbert's 17th problem, except for certain restrictions on the field K Dubois has shown that the theorems of Artin and Pfister are not valid in all fields K Is it possible (by, perhaps, increasing Pfister's bound on the number of needed squares) to relax the present limitations on IK? How far? In particular, what can be said in the case IK = O? Is the conjectured bound 2n + 3 valid? Does there exist, perhaps for a certain class of fields (say, subfields of the reals that are not real closed) a bound of the type 2n + 0(1), or 2n + O(n), or 2n(1 + 0(1))? Here the implied constants may be absolute, or depend on the field. 3. Hilbert showed that, if the function f to be represented by a sum of squares belongs to a ring T, it is, in general necessary to accept members of the field of fractions of T as elements to be squared. On the other hand, in certain cases (see [147J and [39J) this is not necessary and f = 'Dg?, with giET, and with the number of needed squares unchanged. Are there any other such cases? Are there cases in which it is possible to select gi E T by increasing the number of squares? 4. Let IK be a field, IKj = lK(xj, ... ,xn) and Q(xj, ... ,xn)ElK j; under what conditions is Q(x) = a solvable for every 0 =f= a ElK? Is it sufficient to have Q a form of m ~ 2n variables? The natural setting for this problem seems to be IK real closed; is that in fact the case? 220 Appendix: Open Problems 5. As seen, the genus ofL:7=1 xl (over Z) contains a single class for 1 ,,:.;; n < 8; is that true in a more general setting? In particular, what can be said about the genera of a preassigned number of classes, when arbitrary algebraic number fields are considered? This problem is partly solved (see [211], [212], [188], and [233]), and it appears that, at least for n ~ 3, only in finitely many algebraic number fields does the genus contain only one class. 6. In more elaborate presentations of the circle method, one computes sepa­ rately the contributions to the main integral of the major and of the minor arcs. The major arcs consist ofthose points ofthe unit circle that are "close" to points e21cihlk with "small" k. The complements of the major arcs are the minor arcs. The integral along the major arcs leads to the principal term; that along the minor arcs leads to an error term. For precise definitions see [71] or [271]. A really good estimate for the contributions ofthe minor arcs would be an important achievement. References * Books are marked by an asterisk. *1. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. 7th ed., New York: Dover Publications Inc., 1968. *2. L. Ahlfors. Complex Analysis. 2nd ed. New York: McGraw-Hill, 1966. *3. Amer. Math. Soc. Proc. Symposia Pure Math. Vol. 28. Providence, RI, 1976. 4. K. Ananda Rau. On the representation of a number as the sum of an even number of squares, J. Madras Univ. Part B 24 (1954),61-89. *5. G. Andrews. The Theory ofPartitions. Encyclopedia of Mathematics and Its Applications, Vol. 2. Reading, MA: Addison-Wesley Publishing Company, 1976. 6. G. Andrews. Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441-484. 7. N. C. Ankeny. Representations of primes by quadratic forms, Arner. J. Math. 74 (1952), 913-919. 8. N. C. Ankeny. Sums of3 squares, Proc. Arner. Math. Soc. 8 (1957), 316-319. 9. E. Artin. Kennzeichnung des Kiirpers der reelen algebraischen Zahlen, Abh. Sem. Ham­ burg. Univ. 3 (1924), 319-323. 10. E. Artin. Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Sem. Hamburg. Univ. 5 (1926),100-115. II. E. Artin and O. Schreier. Algebraische Konstruktion reeller Kiirper, Abh. Seminar Ham­ burg. Univ. 5 (1926), 85-99. *12. P. Bachmann. Die Arithmetik der Quadratischen Formen. Leipzig: Teubner, 1898. 13. A. Baker. On the class number of imaginary quadratic fields, Bull. Amer. Math. Soc. 77 (1971), 678-684. 14. A. Baker. Imaginary quadratic fields with class-number two, Ann. of Math. (2) 94 (1971), 139-152. 15. A. Baltes, P. K. J. Draxl, and E. R. Hilf. Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, J. Reine Angew. Math. 268/269 (1974), 410-417. 16. R. P. Bambah and S. Chowla. On numbers that can be expressed as a sum of two squares, Proc. Nat. Inst. Sci. India 13 (1947),101-103. 17. D. P. Banerjee. On the application of the congruence property of Ramanujan's Function to certain quaternary forms, Bull. Calcutta Math. Soc. 37 (1945), 24-26. 18. P. T. Bateman. On the representation of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951),70-101. 19. P. T. Bateman. Problem E 2051, Amer. Math. Monthly. Solution by the proposer, Arner. Math Monthly 76 (1969) 190-191. 20. P. T. Bateman and E. Grosswald. Positive integers expressible as a sum of three squares in essentially only one way, J. Number Theory 19 (1984), 301-308. *21. R. Bellman. A BriefIntroduction to Theta Functions. New York: Holt, Rinehart &Winston, 1961. *22. H. Behnke and F. Sommer. Theorie der analytischen Funktionen einer komplexen Ver­ iinderlichen. Grundlehren Math. Wiss. 74. Berlin: Springer-Verlag, 1955. 23. G. Benneton. Sur la representation des nombres entiers par la somme de 2m carres et sa mise en facteurs, C. R. Acad Sci. Paris 212 (1941),591-593,637-639. 24. B. Berndt. The evaluation of character series by contour integration, Publ. Elektrotechn. 222 References Fac. Ser. Mat. i Fiz., No. 381-409 (1972), 25-29. *25. G. Birkhof and S. MacLane. A Survey of Modern Algebra. New York: Macmillan Co., 1947. 26. M. N. Bleicher and M. I. Knopp. Lattice points in a sphere, Acta Arithmetica 10 (1965), 369-376. 27. F. van der Blij. On the theory of simultaneous linear and quadratic representations, Parts I, II, III, IV, V, Nederl. Akad. Wetensch. Proc. 50, 31-40; 41-48, 166-172,298-306, 390-396; Indag. Math. 9(1947), 16-25,26-33, 129-135, 188-196,248-254. *28. Z. I. Borevich and I. R. Safarevich. Number Theory. New York: Academic Press, 1966. 29. H. Braun. Uber die Zerlegung quadrati scher Formen in Quadrate. J. Reine Angew. Math. 178 (1938), 34-64. 30. P. Bronkhorst. On the Number of Solutions of the System of Diophantine Equations x~ + ... + x; = n, XI + ... + Xs = mfor s = 6 and 8 (in Dutch). Thesis, Univ. of Gronin­ gen, 1943. Amsterdam: North Holland Publishing Co., No.6, 1943. 31. D. A. Buell. Small class numbers and extreme values of L-functions of quadratic fields, Math. Compo 31 (1977),786-796. 32. L. Carlitz. On the representation of an integer as the sum of 24 squares, Nederl. Akad. Wetensch. Indag. Math 17 (1955), 504-506. 33. L. Carlitz. Some partition formulas related to sums of squares, Nieuw Arch. Wisk. (3) 3 (1955), 129-133. 34. L. Carlitz. Note on sums of 4 and 6 squares, Proc. Amer. Math. Soc. 8 (1957), 120-124. 35. J. W. S. Cassels. On the representation of rational functions as sums of squares, Acta Arith. 9 (1964), 79-82. 36. J. W. S. Cassels, W. J. Ellison, and A. Pfister. On sums of squares and on elliptic curves over function fields, J. Number Theory 3 (1971),125-149. 37. M. C. Chakrabarti. On the limit points of a function connected with the 3-square problem, Bull. Calcutta Math. Soc. 32 (1940),1-6. 38. Chen, Jin-run. The lattice points in a circle, Sci. Sinica 12 (1963),633-649. 39. D. M. Choi, T. Y. Lam, B. Resnick, and A. Rosenberg. Sums of squares in some integral domains, J. Algebra 65 (1980), 234-256. 40. P. Chowla. On the representation of - I as a sum of two squares of cyclotomic integers, Norske Vid. Selsk. Forh. (Trondheim) 42 (1969), 51-52. See also J. Number Theory I (1968),208-210. 41. P. Chowla and S. Chowla. Determination of the Stufe of certain cyclotomic fields, J. Number Theory 2 (1970),271-272. 42. S. Chowla. An extension of Heilbronn's class-number theorem, Quart. J. Math. Oxford 5 (1934),304-307. *43. H. Cohn. Advanced Number Theory. New York: Dover Publications, 1980. 44. H. Cohn. Numerical study of the representation of a totally positive quadratic integer as the sum of quadratic integral squares, Numer. Math. I (1959), 121-134. 45. H. Cohn. Decomposition into 4 integral squares in the fields of 21/2 and 31/3, Amer.
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