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Diagonal Equations Over Large Finite Fields

Published online by Cambridge University Press:  20 November 2018

Charles Small
Charles Small*
Affiliation:
Queen's University, Kingston, Ontario
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We consider polynomials of the form

with non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.

We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.

It should be noted that in studying diagonal equations

we fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 249 - 262
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Anderson, D., Problem 6201, Amer. Math. Monthly 85 (1978), 203 and 86 (1979), 869–870.Google Scholar
2. Dodson, M. M., Homogeneous additive congruences, Phil. Trans. Roy. Soc. A 261 (1967), 163–210.Google Scholar
3. Dodson, M. M., Some estimates for diagonal equations over -adic fields, Acta Arith. 40 (1982), 117–124.Google Scholar
4. Hua, L. K. and Vandiver, H. S., On the existence of solutions of certain equations in ajinite field, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 258–263.Google Scholar
5. Ireland, K. and Rosen, M., Elements of number theory (Bogden and Quigley, 1972).Google Scholar
6. Joly, J.-R., Equations et variétés algébriques sur un corps fini, Ens. Math. 19 (1973), 1–117.Google Scholar
7. Joly, J.-R., Nombre de solutions de certaines equations diagonales sur un corps fini, C. R. Acad. Sci Paris 272 (1971), 1549–1552.Google Scholar
8. Morlaye, B., Equations diagonales non homogènes sur un corps fini, C. R. Acad. Sci. Paris 272 (1971), 1545–1548.Google Scholar
9. Orzech, M., Forms of low degree and sums of dth powers in finite fields, preprint.Google Scholar
10. Schmidt, W. M., Equations over finite fields an elementary approach, Springer Lecture Notes 536 (1976).CrossRefGoogle Scholar
11. Serre, J.-P., A course in arithmetic (Springer, 1973).CrossRefGoogle Scholar
12. Small, C., Sums of powers in large finite fields, Proc. A. M. S. 65 (1977), 35–36.Google Scholar
13. Tietäväinen, A., On the non-trivial solvability of some equations and systems of equations in finite fields, Ann. Acad. Sci. Fenn. Ser. A, I. 360 (1965), 1–38.Google Scholar
14. Tietäväinen, A., On diagonal forms over finite fields, Ann. Univ. Turku, Ser. A I 118 (1968).Google Scholar
15. Tietäväinen, A., Note on Waring's problem (mod p), Ann. Acad. Sci. Fenn. A I 554 (1973).Google Scholar
16. Tornheim, L., Sums of nth powers infields of prime characteristic, Duke Math. J. 4 (1938), 359–362.Google Scholar
17. Weil, A., Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508.Google Scholar