Additive structure of non-monogenic simplest cubic fields
Authors
Gil-Muñoz, D.; Tinková, M.
Year
2025
Published
The Ramanujan Journal. 2025, 66(3), ISSN 1382-4090.
Type
Article
Departments
Annotation
We consider Shanks’ simplest cubic fields K for which the index [O_K : Z[rho]] of a root rho of the defining parametric polynomial is 3. For them, we study the additive indecomposables of K and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of K. Mainly, we obtain good bounds on the ranks of universal quadratic forms over K and prove that the Pythagoras number of O_K is 6.
Arithmetic of cubic number fields: Jacobi–Perron, Pythagoras, and indecomposables
Authors
Kala, V.; Sgallová, E.; Tinková, M.
Year
2025
Published
Journal of Number Theory. 2025, 273 37-95. ISSN 0022-314X.
Type
Article
Departments
Annotation
We study a new connection between multidimensional continued fractions, such as Jacobi-Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions.
The lifting problem for universal quadratic forms over simplest cubic fields
Authors
Gil-Muñoz, D.; Tinková, M.
Year
2024
Published
Bulletin of the Australian Mathematical Society. 2024, 110(1), 77-89. ISSN 0004-9727.
Type
Article
Departments
Annotation
The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.
On the Pythagoras number of the simplest cubic fields
Type
Article
Departments
Annotation
Let ρ be a root of the polynomial x^3−ax^2−(a+3)x−1 where a≥3. We show that the Pythagoras number of the order Z[ρ] is equal to 6.
Trace and norm of indecomposable integers in cubic orders
Authors
Year
2023
Published
The Ramanujan Journal. 2023, 61(4), 1121-1144. ISSN 1382-4090.
Type
Article
Departments
Annotation
We study the structure of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal traces of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surprising, as in the so-far studied examples of quadratic and simplest cubic fields, this minimum is 1 or 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.