Kessler, Volker On the minimum of the unit lattice. (English) Zbl 0755.11034 Sémin. Théor. Nombres Bordx., Sér. II 3, No. 2, 377-380 (1991). 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: \[ \lambda(L)>{1\over\sqrt{r+1}}{1\over1000}\left({\log\log n\over\log n}\right)^ 3. \] Reviewer: D.W.Boyd (Vancouver) Cited in 3 Documents MSC: 11R27 Units and factorization 11H06 Lattices and convex bodies (number-theoretic aspects) Keywords:lattices; Mahler’s measure; Smyth’s inequality; Dobrowolski’s inequality; unit group Citations:Zbl 0235.12003; Zbl 0416.12001 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.