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On the minimum of the unit lattice. (English) Zbl 0755.11034

Let \(K\) be a finite extension of \(\mathbb{Q}\) of degree \(n\) with maximal order \(R\). As usual, let \(r_ 1\) and \(2r_ 2\) be the number of real and non- real embeddings of \(K\) into \(\mathbb{C}\), and let \(r=r_ 1+r_ 2-1\) be the rank of the unit group \(R^*\). Let Log denote the standard mapping of \(K^*\) into \(\mathbb{R}^ r\) so that \(L:=\text{Log}(R^*)\) is a lattice of rank \(r\). Let \(\lambda(L)\) denote the minimum of the Euclidean norms of the non-zero vectors in \(L\). Using results of C. J. Smyth [Bull. Lond. Math. Soc. 3, 169-175 (1971; Zbl 0235.12003)], and E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)], the author proves the following lower bound: \[ \lambda(L)>{1\over\sqrt{r+1}}{1\over1000}\left({\log\log n\over\log n}\right)^ 3. \]

MSC:

11R27 Units and factorization
11H06 Lattices and convex bodies (number-theoretic aspects)
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References:

[1] Buchmann, ZurKomplexität der Berechnung von Einheiten und Klassenzahlen algebraicher Zahlkörper, HabilitationsschriftDüsseldorf (1987).
[2] Buchmann, Kessler, Computing a reduced lattice basis from a generating system, to appear.
[3] Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta arithmetica34 (1979), 391-401. · Zbl 0416.12001
[4] SchinzelZassenhaus, A refinement of two theorems of Kronecker, Mich. Math. J.12 (1965), 81-84. · Zbl 0128.03402
[5] SmythOn the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc.3 (1971), 169-175. · Zbl 0235.12003
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