56 research outputs found

    Large Algebraic Integers

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    An algebraic integer is said large if all its real or complex embeddings have absolute value larger than 11. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions

    Toric varieties and Gr\"obner bases: the complete Q-factorial case

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    We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors VV as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by VV and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels' arguments on the Gr\"obner fan of toric ideals to our complete case; we give a characterization of the Gr\"obner region and show an explicit correspondence between Gr\"obner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to VV allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by VV. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of \Q-factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.Comment: 18 pages, 2 figures. Final version accepted for publication in the special issue of the Journal AAAECC, concerning "Algebraic Geometry from an Algorithmic point of View

    On periodicity of p-adic Browkin continued fractions

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    The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every n≥1, there exist infinitely many square roots of integers with periodic Browkin expansion of period 2^n, extending a previous result of Bedocchi obtained for n=1
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