Advertisement

Weakly Positive 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 positive unit forms, and their sets of positive roots R+(q). A characterization of critical unit forms, those forms that fail to be weakly positive but all their restrictions are, is presented. We give criteria to identify weakly positive forms, for instance finiteness of the set of positive roots R+(q) (due to Drozd and Happel), and Zeldych’s Theorem (looking for properties of principal submatrices of the symmetric matrix associated to q). Ovsienko’s Theorem is also proved, setting 6 as a bound for the entries of all positive roots of a weakly positive unit form. Those forms with a positive root reaching bound 6 are studied, following Ostermann and Pott. A classification of thin forms due to Dräxler, Drozd, Golovachtchuk, Ovsienko and Zeldych is sketched, as well as a procedure to find all weakly positive unit forms starting with (good) thin forms.

References

  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. 26.
    Gabriel, P., Roiter, A.V.: Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopedia of Mathematical Science, vol. 73. Springer, Berlin, Heidelberg, New York (1992)Google Scholar
  3. 30.
    Happel, D.: The converse of Drozd’s theorem on quadratic forms. Commun. Algebra 23(2), 737–738 (1995)MathSciNetCrossRefGoogle Scholar
  4. 42.
    Ostermann, A., Pott, A.: Schwach positive ganze quadratische Formen, die eine aufrichtife, positive Wurzel mit einem Koeffizienten 6 besitzen. J. Algebra 126, 80–118 (1989)MathSciNetCrossRefGoogle Scholar
  5. 43.
    Ovsienko, A.: Integral weakly positive forms (Russian). In: Schurian Matrix Problems and Quadratic Forms, pp. 3–17. Mathematics Institute of the Academy of Sciences of the Ukrainian SSR, Kiev (1978)Google Scholar
  6. 46.
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, New York (1984)Google Scholar
  7. 52.
    von Höhne, H.-J.: On weakly positive unit forms. Comment. Math. Helvetici 63, 312–336 (1988)MathSciNetCrossRefGoogle Scholar
  8. 53.
    von Höhne, H.-J.: Edge reduction for unit forms. Arch. Math. 65, 300–302 (1995)MathSciNetCrossRefGoogle Scholar
  9. 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