Corvaja zannier 2005 height sunit points

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corollary_1: Corvaja and Zannier's corollary that on a finitely generated subgroup of G_m^2 the height of (u-1)/(v-1) is asymptotic to h(1 : u : v) along multiplicatively independent pairs as max{h(u), h(v)} tends to infinity, which the paper says gives the gcd(a^n-1, b^n-1) bound at once.

main_theorem: Corvaja and Zannier's Main Theorem that for a rational function f whose numerator and denominator monomials include 1, the points of a finitely generated subgroup of G_m^2 where h(f(u, v)) is below (1 - eps) times the largest monomial height have Zariski closure a finite union of translates of proper subtori.

theorem_1: Corvaja and Zannier's theorem that for coprime non-constant p, q not both vanishing at the origin and a finitely generated subgroup of G_m^2, the points where h(p/q) falls short of h(p : q : 1) by eps max{h(u), h(v)} have Zariski closure a finite union of translates of 1-dimensional subtori and a finite set.

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Pietro Corvaja and Umberto Zannier, A lower bound for the height of a rational function at S-unit points, arXiv:math/0311030v2 (2004; published Monatsh. Math. 144 (2005) 203-224; 18 pp.).

The paper generalizes the gcd bounds of Bugeaud, Corvaja and Zannier to lower bounds for heights of rational functions evaluated at points $(u,v)$ of a finitely generated subgroup $\Gamma$ of $\mathbf G_m^2(\overline{\mathbb Q})$, via the Subspace Theorem.

Theorem 1 (p. 1) treats $p/q$ for coprime non-constant $p,q$ not both vanishing at $(0,0)$: for every $\epsilon>0$ the points where $h(p/q)<h(p:q:1)-\epsilon\max\{h(u),h(v)\}$ have Zariski closure a finite union of translates of 1-dimensional subtori, which can be effectively determined, and a finite set. The Main Theorem (p. 2) treats a rational function $f$ whose numerator and denominator monomials include $1$: the points where $h(f(u,v))<(1-\epsilon)\max_ih(T_i(u,v))$ have Zariski closure a finite union of translates of proper subtori, and outside such a union $h(f(u,v))>(1-\epsilon)\max\{h(u)/(2\deg_Yf),h(v)/(2\deg_Xf)\}$.

Corollary 1 (p. 2) gives $h((u-1)/(v-1))\sim h(1:u:v)$ along multiplicatively independent pairs in $\Gamma$ as $\max\{h(u),h(v)\}\to\infty$. The paper says (p. 2) that the main results of its references [1] and [6] are immediate consequences of it; [1] is the bound $\gcd(a^n-1,b^n-1)<\exp(\epsilon n)$ for multiplicatively independent positive integers $a,b$ as $n\to\infty$ (p. 1). It proves Corollary 1 from its Proposition 2 (p. 7), which places all but finitely many exceptions to (1.3) on subgroups $u^p=v^q$ with $p,q$ coprime and $\max\{|p|,|q|\}\le\epsilon^{-1}$.

Pages and labels are those of arXiv:math/0311030v2 (18 pp.), not the journal's pages 203–224.

Source: https://arxiv.org/abs/math/0311030; the copy read for this card is arXiv:math/0311030v2. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0311030), every other right reserved.

Read status: claims checked for Theorem 1, Corollary 1, the Main Theorem, (1.2), (1.3), Lemma 2 and Propositions 1 and 2, read clause by clause on the page images of the print; the proofs of Proposition 2, Theorem 1, Corollary 1 and the Main Theorem followed for structure. Nothing here is independently reviewed. Result pages: theorem_1, corollary_1 and main_theorem.

Bears on. #770: by the paper's remark (p. 2), the fixed-base bound $\gcd(a^n-1,b^n-1)<\exp(\epsilon n)$ for large $n$ is an immediate consequence of Corollary 1; that bound limits the size of a common divisor of two power differences; it does not say that a gcd equals one, and the paper decides none of the problem's three questions.

Results.

• Theorem 1 (p. 1): the solutions in $\Gamma$ of $h(p/q)<h(p:q:1)-\epsilon\max\{h(u),h(v)\}$ have Zariski closure a finite union of translates of 1-dimensional subtori and a finite set.
• Corollary 1 (p. 2): $h((u-1)/(v-1))\sim h(1:u:v)$ for multiplicatively independent $(u,v)\in\Gamma$ as $\max\{h(u),h(v)\}\to\infty$.
• Main Theorem (p. 2): the solutions in $\Gamma$ of $h(f(u,v))<(1-\epsilon)\max_ih(T_i(u,v))$ have Zariski closure a finite union of translates of proper subtori.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.