Weakly Nonnegative Quadratic Forms

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


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.


  1. 22.
    Dräxler, P., Drozd, A., Golovachtchuk, N.S., Ovsienko, S.A., Zeldych, M.V.: Towards the classification of sincere weakly positive unit forms. Eur. J. Combin. 16, 1–16 (1995)MathSciNetCrossRefGoogle Scholar
  2. 23.
    Dräxler, P., Golovachtchuk, N.S., Ovsienko, S.A., de la Peña, J.A.: Coordinates of maximal roots of weakly non-negative unit forms. Colloq. Math. 78(2), 163–193 (1998)MathSciNetCrossRefGoogle Scholar
  3. 31.
    Happel, D., de la Peña, J.A.: Quadratic forms with a maximal sincere root. Can. Math. Soc. 18, 307–315 (1996)MathSciNetzbMATHGoogle Scholar
  4. 44.
    Ovsienko, A.: Maximal roots of sincere weakly nonnegative forms. Lecture at the workshop on Quadratic Forms in the Representation Theory of Finite-Dimensional Algebras, Bielefeld, November 9–12 (1995)Google Scholar
  5. 46.
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, New York (1984)Google Scholar
  6. 54.
    von Höhne, H.-J., de la Peña, J.A.: Isotropic vectors of non-negative integral quadratic forms. Eur. J. Combin. 19, 621–638 (1998)MathSciNetCrossRefGoogle Scholar
  7. 55.
    Zeldych, M.V.: A criterion for weakly positive quadratic forms (Russian). In: Linear Algebra and the Theory of Representations. SSR, Kiev (1983)Google Scholar

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

Personalised recommendations