Explicit uniform estimation of rational points. I: Estimation of heights (Q2904010)

The purpose of these two papers (for part II, cf. [ibid. 668, 89--108 (2012; Zbl 1248.14031)]) is to develop the ``determinant method'' due to \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] and extended by the reviewer [Ann. Math. (2) 155, No. 2, 553--598 (2002; Zbl 1039.11044)]. The primary new idea is to interpret the method within the context of Arakelov geometry, and to use \textit{J.-B. Bost}'s slope theory, see for example [Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 537--562 (2006; Zbl 1141.14011)]. For a number field \(K\) the outcome is a result for a general integral closed subvariety \(X\) of \(\mathbb{P}_K^n\) of dimension \(d\) and degree \(\delta\). Given any \(\varepsilon>0\) take \(D\) to be the smallest integer greater than NEWLINE\[NEWLINE\max\{(\varepsilon^{-1}+1)(2\delta^{-1/d}(d+1)+\delta-2)\,,\, 2(n-d)(\delta-1)+d+2\}.NEWLINE\]NEWLINE Then there is an explicitly computable constant \(C=C(\varepsilon,\delta,n,d,K)\) such that if \(B\geq e^{\varepsilon}\) the rational points of \(X\) of exponential height at most \(B\) are covered by at most NEWLINE\[NEWLINECB^{(1+\varepsilon)\delta^{-1/d}(d+1)}NEWLINE\]NEWLINE hypersurfaces of degree \(\leq D\) not containing \(X\). In particular the degrees which occur are \(O_n(\varepsilon^{-1}\delta)\).NEWLINENEWLINEThe interest of this result is firstly that it applies to general varieties \(X\) rather than merely to hypersurfaces, and secondly that the bound \(D\) on the degrees is completely explicit. \textit{N. Broberg} [J. Reine Angew. Math. 571, 159--178 (2004; Zbl 1053.11027)], had previously given a result for general varieties which depended on the degrees of the forms in the defining ideal. The present paper removes this dependence. In the case of an affine plane curve over the rationals, \textit{Y. Walkowiak} [Acta Arith. 116, No. 4, 343--362 (2005; Zbl 1071.12002)] gives a completely explicit bound, including the constant. Although the present bound is slightly less explicit it is far more general.NEWLINENEWLINEA second elegant result is given for the case in which \(X\) is an integral plane curve. Here it is shown that when \(B=\delta\) the number of rational points of \(X\) of exponential height at most \(\delta\) is \(O_{\varepsilon,K}(\delta^{2+\varepsilon})\). This answers a question posed by the reviewer.NEWLINENEWLINEThe first of the two papers develops the necessary machinery in Arakelov geometry, using Bost's slope method. The results here are potentially of independent interest. It is shown how to represent \(X\) using generators of low degree, via a ``Cayley form'' in the spirit of \textit{F. Catanese} [J. Algebr. Geom. 1, No. 4, 561--595 (1992; Zbl 0807.14006)]. This allows for a suitable description of the singular locus of \(X\). The first paper also considers the geometric and arithmetic Hilbert--Samuel functions, for which good bounds are required.NEWLINENEWLINEThe second paper is devoted to a new interpretation of the determinant method via the slope method. For general number fields \(K\) the new formulation has the advantage (among others) of producing estimates which depend only on the degree of \(K\) and not on its discriminant.