Positive Quadratic Forms

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


An integral quadratic form is a unit form if all its diagonal coefficients are equal to one. In this chapter we study positive unit forms, that is, those integral quadratic unit forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) with q(x) > 0 for any nonzero vector x in \(\mathbb {Z}^n\). A unit form q is critical nonpositive if it is not positive, but each proper restriction of q is. A vector v is called radical for q if q(v + u) = q(u) for any vector u in \(\mathbb {Z}^n\). We prove Ovsienko’s Criterion: a unit form in n ≥ 3 variables is critical nonpositive if and only if q is nonnegative with radical generated by a radical vector with no zero among its entries. One of the most important tools in the theory of integral quadratic forms, inflations and deflations, are introduced in this chapter, and are used to provide a classification of positive unit forms in terms of Dynkin types. A combinatorial characterization of such forms in terms of assemblers of graphs is also presented.


  1. 5.
    Barot, M.: A characterization of positive unit forms. Bol. Soc. Mat. Mexicana (3) 5, 87–93 (1999)Google Scholar
  2. 6.
    Barot, M.: A characterization of positive unit forms, Part II. Bol. Soc. Mat. Mexicana (3) 7, 13–22 (2001)Google Scholar
  3. 7.
    Barot, M., de la Peña, J.A.: The Dynkin type of a non-negative unit form. Expo. Math. 17, 339–348 (1999)MathSciNetzbMATHGoogle Scholar
  4. 18.
    Chartrand, G., Lesniak, L., Zhang, P.: Graphs and Digraphs, 5th edn. Chapman and Hall/CRC Press, Boca Raton, FL (2015)zbMATHGoogle Scholar
  5. 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
  6. 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
  7. 35.
    Jiménez González, J.A.: Incidence graphs and non-negative integral quadratic forms. J. Algebra 513, 208–245 (2018)MathSciNetCrossRefGoogle Scholar
  8. 46.
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, New York (1984)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