Small zeros of additive forms in many variables

Author:
Wolfgang M. Schmidt

Journal:
Trans. Amer. Math. Soc. **248** (1979), 121-133

MSC:
Primary 10B30; Secondary 10J10

MathSciNet review:
521696

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Abstract: It is shown that if *s* is large as a function of *k* and of , then the diophantine equation with positive coefficients , has a nontrivial solution in nonnegative integers , not exceeding , where *m* is the maximum of the coefficients.

**[1]**B. J. Birch,*Small zeros of diagonal forms of odd degree in many variables*, Proc. London Math. Soc. (3)**21**(1970), 12–18. MR**0266855****[2]**H. Davenport,*Analytic methods for Diophantine equations and Diophantine inequalities*, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman; Edited and prepared for publication by T. D. Browning. MR**2152164****[3]**H. Davenport and D. J. Lewis,*Homogeneous additive equations*, Proc. Roy. Soc. Ser. A**274**(1963), 443–460. MR**0153655****[4]**A. O. Gelfond and Yu. V. Linnik,*Elementary methods in analytic number theory*, Translated by Amiel Feinstein. Revised and edited by L. J. Mordell, Rand McNally & Co., Chicago, Ill., 1965. MR**0188135****[5]**U. V. Linnik,*An elementary solution of the problem of Waring by Schnirelman’s method*, Rec. Math. [Mat. Sbornik] N.S.**12(54)**(1943), 225–230 (Russian, with English summary). MR**0009777****[6]**Jane Pitman,*Bounds for solutions of diagonal equations*, Acta Arith.**19**(1971), 223–247. (loose errata). MR**0297701****[7]**Hans Peter Schlickewei,*On indefinite diagonal forms in many variables*, J. Reine Angew. Math.**307/308**(1979), 279–294. MR**534226**, 10.1515/crll.1979.307-308.279

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1979-0521696-3

Article copyright:
© Copyright 1979
American Mathematical Society