Schmidt, Wolfgang M. The density of integer points on homogeneous varieties. (English) Zbl 0561.10010 Acta Math. 154, 243-296 (1985). Let \(\mathcal F\) be a set of \(r\) rational forms of odd degrees \(\leq k\) in \(s\) variables. It was shown by B. J. Birch [Mathematika 4, 102–105 (1957; Zbl 0081.04501)], by the Brauer reduction method, that there exists \(c_1(k,r)\) such that \(\mathcal F\) has a non-trivial rational zero whenever \(s\geq c_1\). The principal object of this paper is to prove this result by the Hardy-Littlewood circle method. In fact it is shown that \(Z_P({\mathcal F})\gg P^{s-c_2}\) for \(s\geq c_2(k,r)\), where \(Z_P(\mathcal F)\) denotes the number of common integer zeros \((x_1,\ldots, x_s)\) of the forms \(\mathcal F\) satisfying \(| x_i| \leq P\). The paper allows estimates for \(c_2(k,r)\) in principle, and these would probably be better than those obtained for \(c_1\) by Birch’s method. However a reasonable bound for \(c_2(5,1)\), say, (i.e. a single quintic form) still seems out of reach, partly because the corresponding \(p\)-adic problem has not been settled for small \(p\). It should be noted that for systems where the singular locus is not too large, the analytic method of B. J. Birch [Proc. R. Soc. Lond., Ser. A 265, 245–263 (1962; Zbl 0103.03102)] applies. In particular, if \(\mathcal F\) consists of a single non-singular form with zeros in every \(p\)-adic field, then we may take \(c_2=(k-1)2^k\). The paper follows H. Davenport [ibid. 272, 285–303 (1963; Zbl 0107.04102)] and H. Davenport and D. J. Lewis [Am. J. Math. 84, 649–665 (1962; Zbl 0118.28103)] in using the invariant \(h(F)\) of the rational form \(F\), defined as the least \(h\) for which one can write \(F=G_1H_1+G_2H_2+\ldots+G_hH_h\), with rational forms \(G_i\), \(H_i\) of positive degree. The paper starts with a fairly routine presentation of the circle method, and proceeds to the estimation of multiple exponential sums via Weyl’s inequality. The final section of the paper, which is the most novel and difficult, relates \(h(F)\) to the density of integer solutions of certain multilinear forms associated with \(F\). These multilinear forms are familiar from Davenport’s work (loc. cit.) on the cubic case, and the kernel of the problem is the extension of his results to higher degrees. In fact a totally different method is required. In addition to the main theorem stated above, other more quantitative results are given. Moreover it is shown that one can allow forms of even degree in \(\mathcal F\) under certain circumstances. Finally an estimate for multiple exponential sums, which arises in the course of the work, is given. This important paper makes the first progress in the application of analytic methods to general systems of forms. It is to be hoped that, in time, the techniques begun here can be refined to yield “realistic” quantitative results. Reviewer: D. R. Heath-Brown (Oxford) Cited in 7 ReviewsCited in 49 Documents MSC: 11E76 Forms of degree higher than two 11L07 Estimates on exponential sums 11P55 Applications of the Hardy-Littlewood method 11D72 Diophantine equations in many variables Keywords:systems of forms; higher degree forms; number of common zeros; rational forms of odd degrees; Hardy-Littlewood circle method; estimation of multiple exponential sums; Weyl’s inequality; multilinear forms; forms of even degree Citations:Zbl 0081.04501; Zbl 0103.03102; Zbl 0107.04102; Zbl 0118.28103 PDFBibTeX XMLCite \textit{W. M. Schmidt}, Acta Math. 154, 243--296 (1985; Zbl 0561.10010) Full Text: DOI References: [1] Birch, B. J., Homogeneous forms of odd degree in a large number of variables.Mathematika, 4 (1957), 102–105. · Zbl 0081.04501 [2] –, Forms in many variables.Proc. Roy. Soc. Ser. A, 265 (1962), 245–263. · Zbl 0103.03102 [3] Davenport, H., Cubic forms in 32 variables.Philos. Trans. Roy. Soc. London Ser. A, 251 (1959), 193–232. · Zbl 0084.27202 [4] –, Cubic forms in 16 variables.Proc. Roy. Soc. Ser. A, 272 (1963), 285–303. · Zbl 0107.04102 [5] Davenport, H. &Lewis, D. J., Homogeneous additive equations.Proc. Roy. Soc. Ser. A, 274 (1963), 443–460. · Zbl 0118.28002 [6] Lachaud, G., Une presentation adelique de la serie singuliere et du probleme de Waring.Enseign. Math. To appear. · Zbl 0444.10042 [7] Lang, S.,Introduction to algebraic geometry. Interscience Tracts in Pure and Applied Math., 1958. · Zbl 0095.15301 [8] Leep, D. &Schmidt, W. M., Systems of homogeneous equations.Inventiones Math., 71 (1983), 539–549. · Zbl 0504.10010 [9] Schmidt, W. M., Simultaneous rational zeros of quadratic forms.Seminar Delange-Pisot-Poitou 1981. Progress in Math., Vol. 22 (1982), 281–307. [10] –, On cubic polynomials II. Multiple exponential sums.Monatsh. Math., 93 (1982), 141–168. · Zbl 0471.10029 [11] –, On cubic polynomials III. Systems ofp-adic equations.Monatsh. Math., 93 (1982), 211–223. · Zbl 0473.10017 [12] –, On cubic polynomials IV. Systems of rational equations.Monatsh. Math., 93 (1982), 329–348. · Zbl 0481.10015 [13] Seidenberg, A., Constructions in algebra.Trans. Amer. Math. Soc., 197 (1974), 273–313. · Zbl 0356.13007 [14] Tartakovsky, W., Über asymptotische Gesetze der allgemeinen Diophantischen Analyse mit vielen Unbekannten.Bull. Acad. Sci. USSR, (1935), 483–524. · JFM 61.1068.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.