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Heights of algebraic points lying on curves or hypersurfaces. (English) Zbl 0867.11046
Define the absolute logarithmic height of a non-zero algebraic number $$x$$ of degree $$r$$ by $h(x)= {1\over r}\log\Biggl(|a_0|\prod^r_{i=1}\max (1,|x^{(i)}|)\Biggr),$ where $$a_0\prod^r_{i=1} (X-x^{(i)})$$ is the minimal polynomial of $$x$$ in $$\mathbb{Z}[X]$$. Using techniques from Arakelov theory, S. Zhang [Ann. Math., II. Ser. 136, 569-587 (1992; Zbl 0788.14017)] showed that if $$x$$, $$y$$ are algebraic numbers, not equal to 0 or a cube root of unity, with $$x+y+1=0$$, then $$\max(h(x),h(y))\geq c$$, where $$c>0$$ is an absolute constant. Later, D. Zagier [Math. Comput. 61, 485-491 (1993; Zbl 0786.11063)] gave an elementary proof of this fact. In the present paper, the author generalizes this as follows, also with an elementary proof:
Let $$F(X_1,\dots,X_n)\in\mathbb{Z}[X_1,\dots,X_n]$$ be a non-zero polynomial of total degree $$f$$ and coefficients with absolute values at most $$H$$. Then if $$x_1,\dots,x_n$$ are non-zero algebraic numbers with $$F(x_1,\dots,x_n)=0$$, $$F(x^{-1}_1,\dots,x^{-1}_n)\neq 0$$, then $\sum^n_{i=1} h(x_i)\geq 2^{-4f-2n}H^{-1}.$ Independently, F. Beukers and D. Zagier [Preprint 943, Univ. Utrecht] obtained a similar result.
As an application, the author estimates the number of points of bounded height lying on an algebraic plane curve and with coordinates belonging to a given multiplicative group of finite rank. Let $$P(X,Y)\in C[X,Y]$$ be a polynomial not divisible by $$X$$ or $$Y$$. Letting $$a$$, $$b$$ denote the degrees of $$P$$ in $$X$$, $$Y$$, respectively, one defines $$\widetilde P(X,Y)= X^aY^bP(X^{-1},Y^{-1})$$. The polynomial $$P$$ is called reflexive if in $$P$$ and $$\widetilde P$$ the same monomials occur; i.e., for some finite set $$M$$ we have $$P=\sum_{(i,j)\in M}u_{ij}X^iY^j$$, $$\widetilde P=\sum_{(i,j)\in M}\widetilde u_{ij}X^iY^j$$ with $$u_{ij}\neq 0$$, $$\widetilde u_{ij}\neq 0$$. Now suppose that $$P$$ is irreducible in $$C[X,Y]$$ and not reflexive, that $$P$$ has total degree $$p$$ and that there are precisely $$m$$ monomials occurring in $$P$$. Further, let $$\Gamma$$ be a subgroup of rank $$r$$ of $$(\overline Q^\times)^2$$. Then $$P$$ has at most $$(2^{9pm})^{r+1}C^r$$ zeros $$(x,y)$$ with $$(x,y)\in\Gamma$$ and $$h(x)+ h(y)\leq C$$.

##### MSC:
 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14H25 Arithmetic ground fields for curves
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##### References:
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