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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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