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On the Number and Distribution of Simultaneous Solutions to Diagonal Congruences

Published online by Cambridge University Press:  20 November 2018

Kenneth W. Spackman
Kenneth W. Spackman*
Affiliation:
University of Kentucky, Lexington, Kentucky
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Two aspects, and their connections, of the problem of enumerating solutions to certain systems of congruences are explored in this paper. Although slightly more general cases are mentioned, the basic object of study is a system of diagonal equations

where d1, d2, …, dt are positive integers and the coefficient matrix [aij] has entries from Fp = GF(p), and for which solutions x = (x1, x2 …, xt) ∈ Fpt are sought. Speaking loosely, such a system usually has approximately pt–n solutions in the sense that the difference between pt–n and the correct value becomes small in comparison with pt–n as p becomes large. A parameter is introduced which measures the extent to which the matrix [aij] is non-singular over Fp. The effect of this parameter on the size of the error term (the difference between the number of solutions to (1) and pt–n) is the first aspect of the enumeration problem to be treated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 421 - 436
Copyright
Copyright © Canadian Mathematical Society 1981

References

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