Related papers: Lifting problem for universal quadratic forms over…

We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that $\mathbb Q(\sqrt 5)$ is the only such real quadratic field, and that among fields of degrees 3, 4, 5, and 7…

The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given…

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

For all positive integers $k$ and $N$ we prove that there are infinitely many totally real multiquadratic fields $K$ of degree $2^k$ over $\mathbb Q$ such that each universal quadratic form over $K$ has at least $N$ variables.

We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully…

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…

We show that if $K$ 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 $K$. In particular, we extend…

Given any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms in M variables.

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd…

For any positive integer M we show that there are infinitely many real quadratic fields that do not admit M-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

We prove that there are at most 13 real quadratic fields that admit a ternary universal quadratic lattice, thus establishing a strong version of Kitaoka's Conjecture for quadratic fields. More generally, we obtain explicit upper bounds on…

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a new theorem on the number…

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without…

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of…

We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In…

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.