Recent zbMATH articles in MSC 11G50https://www.zbmath.org/atom/cc/11G502021-04-16T16:22:00+00:00WerkzeugCanonical heights and preperiodic points for certain weighted homogeneous families of polynomials.https://www.zbmath.org/1456.371082021-04-16T16:22:00+00:00"Ingram, Patrick"https://www.zbmath.org/authors/?q=ai:ingram.patrickSummary: A family \(f\) of polynomials over a number field \(K \) will be called weighted homogeneous if and only if \(f_t(z) = F(z^e, t)\) for some binary homogeneous form \(F(X, Y)\) and some integer \(e \geq 2\). For example, the family \(z^d + t\) is weighted homogeneous. We prove a lower bound on the canonical height, of the form
\[
\hat{h}_{f_t}(z)\geq \varepsilon \max \left\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\right\},
\]
for values \(z \in K\) which are not preperiodic for \(f_t\). Here \(\varepsilon\) depends only on the number field \(K\), the family \(f\), and the number of places at which \(f_t\) has bad reduction. For suitably generic morphisms \( \varphi :\mathbb{P}^1 \to \mathbb{P}^1\), we also prove an absolute bound of this form for \(t\) in the image of \(\varphi\) over \(K\) (assuming the \(abc\) Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).On the degeneracy of integral points and entire curves in the complement of nef effective divisors.https://www.zbmath.org/1456.111192021-04-16T16:22:00+00:00"Heier, Gordon"https://www.zbmath.org/authors/?q=ai:heier.gordon"Levin, Aaron"https://www.zbmath.org/authors/?q=ai:levin.aaronSummary: As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.The Lind-Lehmer constant for \(\mathbb{Z}_2^r\times\mathbb{Z}_4^s\).https://www.zbmath.org/1456.112042021-04-16T16:22:00+00:00"Mossinghoff, Michael J."https://www.zbmath.org/authors/?q=ai:mossinghoff.michael-j"Pigno, Vincent"https://www.zbmath.org/authors/?q=ai:pigno.vincent"Pinner, Christopher"https://www.zbmath.org/authors/?q=ai:pinner.christopher-gFor a finite abelian group \(G=\mathbb{Z}_{n_{1}}\times \cdots \times \mathbb{Z}_{n_{k}}\), where \(\mathbb{Z}_{n_{j}}\) \((1\leq j\leq k)\) denotes the cyclic group with order \(n_{j}\), define
\[
\lambda (G)=\min \left( \left\{ \prod_{j_{1}=1}^{n_{1}}\dots\prod_{j_{k}=1}^{n_{k}}\left\vert F(e^{i2\pi j_{1}/n_{1}},\dots,e^{i2\pi j_{k}/n_{k}})\right\vert \mid F\in \mathbb{Z}[x_{1},\dots,x_{k}]\right\} \cap \lbrack 2,\infty )\right) .
\]
According to [\textit{D. Lind} et al., Proc. Am. Math. Soc. 133, No. 5, 1411--1416 (2005; Zbl 1056.43005); \textit{D. Desilva} and \textit{C. Pinner}, Proc. Am. Math. Soc. 142, No. 6, 1935--1941 (2014; Zbl 1294.11185)], if \(G\neq \mathbb{Z}_{2}\), then
\[
\lambda (G)\leq \operatorname{card}(G)-1, \tag{*}
\]
\((\lambda (\mathbb{Z}_{p^{n}}),\lambda (\mathbb{Z}_{2^{n}}))=(2,3)\) for any natural number \(n\) and any odd prime \(p\), and (*) is sharp when \(G=\mathbb{Z}_{3}^{n}\), or when \(G=\mathbb{Z}_{2}^{n}\) and \(n\geq 2\).
In the paper under review, the authors continue to investigate the values of \(\lambda (G)\) for \(G\) running through certain families of \(p\)-groups, where \(p\in \{2,3\}\). Mainly, they show that \(\lambda (\mathbb{Z}_{3}\times \mathbb{Z}_{3^{n}})=8\), \(n\geq 3\Rightarrow \lambda (\mathbb{Z}_{2}\times \mathbb{Z}_{2^{n}})=9\), and equality occurs (again) in (*) whenever \(G\neq \mathbb{Z}_{2}\) and the factors of \(G\) are all \(\mathbb{Z}_{2}\) or \(\mathbb{Z}_{4}\).
The proofs of these results are based on a generalization of Lemma 2.1 of the last mentioned reference about a congruence satisfied by the rational integers defining \(\lambda (G)\), when \(G\) is a \(p\)-group.
Reviewer: Toufik Zaïmi (Riyadh)