A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field

  • Pavlo Yatsyna

    Royal Holloway University of London, Egham, UK
A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field cover
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Abstract

We show that if is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over . In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.

Cite this article

Pavlo Yatsyna, A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field. Comment. Math. Helv. 94 (2019), no. 2, pp. 221–239

DOI 10.4171/CMH/459