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1. CJM 2016 (vol 68 pp. 625)

Ingram, Patrick
 Rigidity and Height Bounds for Certain Post-critically Finite Endomorphisms of $\mathbb P^N$ The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of $f$, has algebraic support. In the case $N=1$, a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible LattÃ¨s maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$. Keywords:post-critically finite, arithmetic dynamics, heightsCategories:37P15, 32H50, 37P30

2. CJM 2007 (vol 59 pp. 1284)

Fukshansky, Lenny
 On Effective Witt Decomposition and the Cartan--Dieudonn{Ã© Theorem Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$. Keywords:quadratic form, heightsCategories:11E12, 15A63, 11G50
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