Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T00:41:58.451Z Has data issue: false hasContentIssue false
  • English
  • Français

On systems of diagonal forms

Part of: Diophantine equations Forms and linear algebraic groups

Published online by Cambridge University Press:  09 April 2009

Michael P. Knapp
Michael P. Knapp
Affiliation:
Mathematical Sciences Department Loyola College4501 North Charles Street Baltimore, MD 21210-2699USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Qp provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Qp. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

MSC classification

Secondary: 11D72: Equations in many variables 11E76: Forms of degree higher than two 11E95: $p$-adic theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 221 - 236
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Ax, J. and Kochen, S., ‘Diophantine problems over local fields I’, Amer. J. Math. 87 (1965), 605–630.CrossRefGoogle Scholar
[2]Bhargava, M., ‘P-orderings and polynomial functions on arbitrary subsets of Dedekind rings’, J. Reine Angew. Math. 490 (1997), 101–127.Google Scholar
[3]Bhargava, M., ‘Generalized factorials and fixde divisors over subsets of a Dedekind domain’, J. Number Theory 72 (1998), 67–75.CrossRefGoogle Scholar
[4]Bhargava, M., ‘The factorial functions and generalizations’, Amer. Math. Monthly 107 (2000), 783–799.CrossRefGoogle Scholar
[5]Davenport, H. and Lewis, D. J., ‘Homogeneous additive equations’, Proc. Roy. Soc. Ser. A 274 (1963), 443–460.Google Scholar
[6]Davenport, H. and Lewis, D. J., ‘Simultaneous equations of additive type’, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557–595.Google Scholar
[7]Knapp, M., ‘Diagonal equations of different degrees over p-adic fields’, Acta Arith. 126 (2007), 139–154.CrossRefGoogle Scholar
[8]Lewis, D. J. and Montgomery, H. L., ‘On zeroes of p-adic forms’, Michigan Math. J. 30 (1983), 83–87.CrossRefGoogle Scholar
[9]Schanuel, S. H., ‘An extension of Chevalley's theorem to congruences modulo prime powers’, J. Number Theory 6 (1974), 284–290.CrossRefGoogle Scholar
[10]Wooley, T. D., ‘On simultaneous additive equations III’, Mathematika 37 (1990), 85–96.CrossRefGoogle Scholar
[11]Wooley, T. D., ‘On simultaneous additive equations I’, Proc. London Math. Soc. (3) 63 (1991), 1–34.Google Scholar