Counting points in a small box on varieties
The number of solutions of bounded size (or more generally, in a box) of a system of congruences \[ G_ 1(x)\equiv G_ 2(x)\equiv...\equiv G_ s(x)\equiv 0 mod p \] (G\({}_ i\) homogeneous polynomials with integral coefficients in n variables) is estimated in both directions, showing that the solutions are uniformly distributed. The validity of the estimate depends on p and the box being large enough, but does not need n to be large [this was the case in previous results of \textit{W. M. Schmidt}, Diophantine Analysis, Lond. Math. Soc. Lect. Note Ser. 109, 37- 66 (1986; Zbl 0589.10024)]. It also depends on a nonsingularity condition for the polynomials in the pencil of \(G_ 1,...,G_ s\). This condition, which previously was assumed to be natural, is shown to be true only rarely if \(s\geq 3.\) The paper gives only an outline of the proofs, postponing details to a later article (see the following review).
scientific article; zbMATH DE number 4095582
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