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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 \(h(x)+ h(y)\leq C\).

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H25 Arithmetic ground fields for curves
Full Text: DOI
[1] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391 – 401. · Zbl 0416.12001
[2] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. Math. 34 (2) (1933), 461–479. · Zbl 0007.19904
[3] H. P. Schlickewei, Equations \(ax+by = 1\), Annals of Math., (to appear).
[4] H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, In preparation. · Zbl 0803.11010
[5] H. P. Schlickewei and E. Wirsing, Lower bounds for the heights of solutions of linear equations, Invent. Math, (to appear). · Zbl 0883.11013
[6] Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. · Zbl 0421.10019
[7] D. Zagier, Algebraic numbers close to both 0 and 1, Math. Comp. 61 (1993), no. 203, 485 – 491. · Zbl 0786.11063
[8] Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569 – 587. · Zbl 0788.14017 · doi:10.2307/2946601 · doi.org
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