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Weakly Nonnegative Quadratic Forms

  • Michael Barot
  • Jesús Arturo Jiménez González
  • José-Antonio de la Peña
Chapter
Part of the Algebra and Applications book series (AA, volume 25)

Abstract

In this chapter we analyze integral quadratic forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) satisfying q(x) ≥ 0 for any positive vector x in \(\mathbb {Z}^n\), so-called weakly nonnegative semi-unit forms. Here a prominent role is played by maximal and locally maximal positive roots of q, which can be used to characterize weak nonnegativity. We also describe hypercritical semi-unit forms, those forms not weakly nonnegative such that any proper restriction is weakly nonnegative. Diverse criteria for weak nonnegativity are provided, including Zeldych’s Theorem and a few algorithms using iterated edge reductions, following von Höhne and de la Peña. A generalization of Ovsienko’s Theorem due to Dräxler, Golovachtchuk, Ovsienko and de la Peña is proved in the last section, for which Ringel’s concepts of graphical and semi-graphical forms are essential.

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michael Barot
    • 1
  • Jesús Arturo Jiménez González
    • 2
  • José-Antonio de la Peña
    • 3
  1. 1.Kantonsschule SchaffhausenSchaffhausenSwitzerland
  2. 2.Instituto de MatemáticasUNAMMexico CityMexico
  3. 3.Instituto de MatemáticasMiembro de El Colegio NacionalUNAMMexico

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