Francesca Aicardi

Publications (2)1 Total impact

• Article: Symmetries of quadratic form classes and of quadratic surd continued fractions. Part II: Classification of the periods’ palindromes

  Francesca Aicardi
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  ABSTRACT: According to a theorem by Lagrange, the continued fractions of quadratic surds are periodic. Their periods may have different types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows classifying the periods of quadratic surds and simultaneously finding the symmetry type of the class of an arbitrary indefinite quadratic form and the number of its integer points contained in each domain of the Poincaré tiling of the de Sitter world, introduced in Part I of this paper. Moreover, we obtain the same result for every class of forms representing zero, i.e., when the quadratic surds are replaced by rational, using the finite continued fraction obtained from a special representative of that class. Finally, we show the relation between the reduction procedure for indefinite quadratic forms defined by continued fractions and the classical reduction theory, which acquires a geometric description by the results in Part I. Keywordscontinued fractions-quadratic forms-reduction theory Mathematical subject classification11A55-11H55
  Bulletin Brazilian Mathematical Society 04/2012; 41(1):83-124. · 0.50 Impact Factor
• Article: Symmetries of quadratic form classes and of quadratic surd continued fractions. Part I: A Poincaré tiling of the de Sitter world

  Francesca Aicardi
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  ABSTRACT: The problem of classifying the indefinite binary quadratic forms with integer coefficients is solved by introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Under the action of the special linear group acting on the integer plane lattice, each class of indefinite forms has a well-defined finite number of representatives inside each such domain. In the second part, we will show how to obtain the symmetry type of a class and also the number of its points in all domains from a single representative of that class.
  Bulletin Brazilian Mathematical Society 04/2012; 40(3):301-340. · 0.50 Impact Factor