On the distribution of solutions to polynomial congruences
After having defined a system of congruences modulo a prime \(p\) (in the family \(\mathfrak F_{m,n}\) of \(n\) polynomials in \(m \geq n+1\) variables, with integer coefficients) and its set of solutions \(\chi_p\), the paper clarifies the properties of a well-shaped domain \(\Omega \in [0,1]^m\) and denotes by \(T_p(\Omega)\) the number of points falling inside of \(\Omega\). Then the paper establishes the following asymptotic formula for the number of solutions in general domains: \[ T_p(\Omega) = \# \chi_p (\mu (\Omega) + O(p^{-1/2(n+1)} \log p)), \] where \( \# \chi_p = T_p([0,1]^m)\) and \(\mu\) is the Lebesgue measure on \([0,1]^m\). It extends the result of \textit{É. Fouvry} [Isr. J. Math. 120, Part A, 81--96 (2000; Zbl 1010.11045)] valid in small cubic boxes only. The proof employs a combinatorial approximation of \(\Omega \) supplied by \textit{W. M. Schmidt} [Acta Arith. 27, 385--396 (1975; Zbl 0274.10038)], an equation about well-shaped domains given by \textit{H. Weyl} [Am. J. Math. 61, 461--472 (1939; Zbl 0021.35503)] and a theorem on the number of points of varieties in finite fields provided by \textit{S. Lang} and \textit{A. Weil} [Am. J. Math. 76, 819--827 (1954; Zbl 0058.27202)]. The author remarks the effectiveness of his approach, non-standard with respect to others (e.g., by \textit{A. Granville} et al. [J. Number Theory 112, No. 2, 216--237 (2005; Zbl 1068.11043)]), and, on the basis of an achievement from \textit{A. Ayyad} et al. [J. Number Theory 59, No. 2, 398--413 (1996; Zbl 0869.11003)], he suggests a combination with an estimate found by the same \textit{I. E. Shparlinski} [Proc. Japan Acad., Ser. A 83, No. 2, 5--9 (2007; Zbl 1123.11026)]. In order to enrich the paper's argumentations, the author recalls a vast literature dealing with the number and distribution of solutions for systems of congruences like, e.g., by \textit{E. Fouvry} and \textit{N. Katz} [J. Reine Angew. Math. 540, 115--166 (2001; Zbl 0986.11054)], by \textit{Wenzhi Luo} [Int. Math. Res. Not. 1999, No.16, 901--907 (1999; Zbl 0969.11026)], by \textit{A. N. Skorobogatov} [Isr. J. Math. 80, No. 3, 359--379 (1992; Zbl 0774.11045)], by \textit{A. Zumalacárregui} [Int. J. Number Theory 7, No. 7, 1835--1839 (2011; Zbl 1250.11036)] and by himself both alone (\textit{I. E. Shparlinski} [Math. Z. 263, No. 3, 619--631 (2009; Zbl 1269.11068)], [J. Number Theory 132, No. 1, 254--257 (2012; Zbl 1268.11051)]) and in collaboration [\textit{M.-C. Chang, J. Cilleruelo, M. Z. Garaev, J. A. Hernández, I. E. Shparlinski} and \textit{A. Zumalacárregui}, ``Concentration of points and isomorphism classes of hyperelliptic curves over a finite field in some thin families'', \url{arXiv:1111.1543v2}], \textit{J. Cilleruelo} et al. [Math. Z. 272, No. 3--4, 825--837 (2012; Zbl 1285.11027)], \textit{I. E. Shparlinski} and \textit{A. N. Skorobogatov} [Mathematika 37, No. 2, 201--208 (1990; Zbl 0719.11086)].
scientific article; zbMATH DE number 6125029