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A Note on Simultaneous Congruences

✍ Scribed by Trevor D. Wooley

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
466 KB
Volume
58
Category
Article
ISSN
0022-314X

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✦ Synopsis

We prove, by an essentially elementary argument, that the number of nonsingular solutions of a system of d simultaneous congruences, to a prime power modulus, in d variables is at most the product of the degrees of the polynomials defining the congruences.

1996 Academic Press, Inc.

where S s, k (n) denotes the usual singular series in Waring's problem (see Vaughan [5, Sect. 2.6]). Then it transpires that the results of this note are required to obtain Ford's new bound G (k) (1+o(1)) k 2 log k, which improves the previous bound G (k) (2+o(1)) k 2 log k due to Wooley [6, Corollary 1.2]. Since the above result on congruences had also article no.

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## Abstract In this paper we establish the (1, 2)‐ and (2, 1)‐transfer principles for finitely decidable locally finite varieties, where a class of structures is finitely decidable if the first order theory of its finite members is recursive. The transfer principles deal with the local structure of