## Lower bounds for heights on finitely generated groups.(English)Zbl 0973.11067

Let $$K$$ be an algebraic number field and $$G$$ a finitely generated subgroup of the multiplicative group $$K^*$$. Suppose that the free part of $$G$$ has rank $$r$$. Let $$h(\alpha)=\prod_{v\in M_K}\max(1,|\alpha|_v)$$ the absolute Weil height of $$\alpha\in K$$, where $$|\cdot|_v$$ $$(v\in M_K)$$ are normalised absolute values satisfying the product formula. Previously, the author and W. M. Schmidt showed that $$G$$ (that is, the free part of $$G$$) has a basis $$\alpha_1,...,\alpha_r$$ such that if $$\alpha$$ is an arbitrary element of $$G$$, expressed uniquely as $$\xi\alpha_1^{x_1}\cdots\alpha_r^{x_r}$$ with $$\xi$$ a root of unity and $$x_1,...,x_r\in\mathbb{Z}$$, then $$h(\alpha)\geq\max_{1\leq i\leq r}\{h(\alpha_i^{x_i})\}^{4^{-r^3}}$$ [Math. Ann. 296, No. 2, 339-361 (1993; Zbl 0805.11029)]. In this paper, the author improves this to (*) $$h(\alpha)\geq\max_{1\leq i\leq r}\{h(\alpha_i^{x_i})\}^{4^{-r}}$$. The proof uses properties of the height such as $$h(\alpha)>1$$ if $$\alpha$$ is not a root of unity, $$h(\alpha^n)=h(\alpha)^{|n|}$$ for $$n\in\mathbb{Z}$$, and $$h(\alpha\beta)\leq h(\alpha)h(\beta)$$. Pick any basis $$\beta_1,...,\beta_r$$ of $$G$$, and define $$\omega({\mathbf x})=\log h(\beta_1^{x_1}\cdots\beta_r^{x_r})$$ for $${\mathbf x}\in\mathbb{Z}^r$$. Then from the above properties of $$h$$ it follows that $$\omega$$ can be extended to a vector norm (i.e., distance function) on $$\mathbb{R}^r$$. Such a vector norm can be approximated by a vector norm on $$\mathbb{R}^r$$ of the shape $$\omega(x)=\sqrt{Q(x)}$$, where $$Q$$ is a positive definite quadratic form on $$\mathbb{R}^r$$. The author proves his theorem by taking a basis $${\mathbf a}_1=(a_{11},...,a_{1r}),...,{\mathbf a}_r$$ of $$\mathbb{Z}^r$$ which is reduced with respect to $$Q$$ in the sense of A. K. Lenstra, H. W. Lenstra jun. and L. Lovász [Math. Ann. 261, 515-534 (1982; Zbl 0488.12001)] and then showing that $$\alpha_i=\beta_1^{a_{i1}}\cdots\beta_r^{a_{ir}}$$ $$(i=1,...,r)$$ is a basis of $$G$$ satisfying (*).

### MSC:

 11G50 Heights 11H06 Lattices and convex bodies (number-theoretic aspects)

### Keywords:

heights; lower bound

### Citations:

Zbl 0805.11029; Zbl 0488.12001
Full Text:

### References:

  Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith.34, 391-404 (1979). · Zbl 0416.12001  John, F.: Extremum problems with inequalites as subsidiary conditions. In: Studies and Essays presented to R. Courant, pp. 187-204. New York: Interscience. 1948. · Zbl 0034.10503  Lenstra, A. K., Lenstra, H. W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann.261, 515-534 (1982). · Zbl 0488.12001  Schlickewei, H. P., Schmidt, W. M.: On polynomial-exponential equations. Math. Ann.296, 339-361 (1993). · Zbl 0805.11029
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