Small points on subvarieties of a torus
The authors improve earlier estimates by \textit{W. M. Schmidt} [Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 235, 157--187 (1996; Zbl 0917.11023)], \textit{S. David} and \textit{P. Philippon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, No.3, 489--543 (1999; Zbl 1002.11055); Errata ibid. 29, 729--731 (2000)], and \textit{F. Amoroso} and \textit{S. David} [Compos. Math. 142, No. 3, 551--562 (2006; Zbl 1116.11045)] on lower bounds for the heights of points on subvarieties of linear tori. The most involved tools used by the authors are upper and lower bounds for the Hilbert function, due to Chardin and Philippon. Apart from that, the authors' proof is much simpler than the arguments of David and Philippon and Amoroso and David. The following notation is used. We view \({\mathbb G}_{{\mathbf m}}^n\) as a subset of \({\mathbb P}^n\) by identifying \((x_1,\ldots ,x_n)\) with \((1:x_1:\cdots :x_n)\). Thus, a subvariety of \({\mathbb G}_{{\mathbf m}}^n\) is the intersection of a subvariety of \({\mathbb P}^n\) with \({\mathbb G}_{{\mathbf m}}^n\). The height \(h\) on \({\mathbb G}_{{\mathbf m}}^n(\overline{{\mathbb Q}})\) is the absolute logarithmic Weil height from \({\mathbb P}^n({{\mathbb Q}})\). Given a subset \(S\) of \({\mathbb G}_{{\mathbf m}}^n(\overline{{\mathbb Q}})\), we denote by \(S(\varepsilon )\) the set of points \({\mathbf x}\in S\) with \(h({\mathbf x})\leq\varepsilon \) and by \(\overline{S(\varepsilon )}\) the Zariski closure in \({\mathbb G}_{{\mathbf m}}^n\) of this set. Given a subvariety \(V\) of \({\mathbb G}_{{\mathbf m}}^n\), we denote by \(\delta (V)\) the minimum of all integers \(\delta\) such that \(V\) is the intersection of hypersurfaces of degree at most \(\delta\). Further, we denote by \(\delta_0(V)\) the minimum of all integers \(\delta\) with the property that there exists an intersection \(X\) of hypersurfaces of degree at most \(\delta\) such that each irreducible component of \(V\) is an irreducible component of \(X\). We restrict ourselves to stating the following result. \textbf{Theorem.} \textit{Let \(V\) be a not necessarily irreducible subvariety of \({\mathbb G}_{{\mathbf m}}^n\) of codimension \(k\), defined over \(\overline{{\mathbb Q}}\). Suppose that \(V=X_k\cup\cdots X_n\), where \(X_j\) is a union of irreducible components of codimension \(j\). Define \[ \theta =\theta (V):= \delta (V)\big( 200n^5\log (n^2\delta (V))\big)^{(n-k)n(n-1)}. \] Then \( \overline{V(\theta^{-1})}=G_k\cup\cdots\cup G_n , \) where \(G_j\) is either the empty set or a finite union of translates of subtori of \({\mathbf G}_{{\mathbf m}}^n\) of codimension \(j\) with \(\delta_0(G_j)\leq\theta\). Moreover, for \(r=k,\ldots ,n\) we have} \[ \sum_{i=k}^r \theta^{r-i}\deg G_i\leq \sum_{i=k}^r \theta^{r-i}\deg X_i\leq\theta^r . \] For instance, let \(V^0\) be \(V\) minus the union of all positive dimensional translates of subtori contained in \(V\). Then the Theorem implies that \[ \#\{ {\mathbf x}\in V^0(\overline{{\mathbf Q}}):\, h({\mathbf x})\leq\theta\}\, \leq \theta^n.\tag{\(*\)} \] . Amoroso and David proved a similar result, but with instead of the above defined \(\theta\) an expression of the shape \(\theta '=\delta (V)(\log \delta (V))^{\lambda (n)}\) where \(\lambda (n)\) is exponential in \(n\). Another improvement concerns a lower bound for the essential minimum \(\hat{\mu}^{\text{ess}}(V)\), of \(V\) which is the supremum of all \(\varepsilon\) such that \(V\varepsilon )\) is not Zariski dense in \(V\). The authors obtained \(\hat{\mu}^{\text{ess}}(V)\geq \delta (V)^{-1}(\log \delta (V))^{-a(n)}\) with \(a(n)\) polynomial in \(n\), whereas Amoroso and David had such a bound with an exponential function \(a(n)\). As an application, the authors give an improvement of an upper bound of Schlickewei, Schmidt and the reviewer for the number of non-degenerate solutions of the equation \(a_1x_1+\cdots +a_nx_n=1\) in \(x_1,\ldots ,x_n\in\Gamma\), where \(a_1,\ldots ,a_n\) are non-zero elements of a field \(K\) of characteristic \(0\), and \(\Gamma\) is a subgroup of \((K^*)^n\) of rank at most \(r\). Schlickewei, Schmidt and the reviewer obtained an upper bound \(c(n,r)=\exp \Big( (6n)^{3n}(r+1)\Big)\). Their argument uses both the quantitative Subspace Theorem to deal with the ``large'' solutions, and a weaker version of (*) due to Schmidt to deal with the ``small'' solutions. This last result of Schmidt was responsible for the doubly exponential dependence on \(n\) of \(c(n,r)\). With their new estimate (\(*\)), the authors manage to improve \(c(n,r)\) to \((8n)^{4n^4(n+r+1)}\).
scientific article; zbMATH DE number 5688238
scientific article