Sums of two units in number fields
Autoři
Tinková, M.; Visser, R.; Yatsyna, P.
Rok
2026
Publikováno
Mathematische Zeitschrift. 2026, 312(2), ISSN 0025-5874.
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Let K be a number field with ring of integers \mathcal{O}_K. Let \mathcal{N}_K be the set of positive integers n such that there exist units \varepsilon, \delta \in \mathcal{O}_K^\times satisfying \varepsilon + \delta = n. We show that \mathcal{N}_K is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation \varepsilon + \delta = n when K is either cyclic or has negative discriminant.
Additive structure of non-monogenic simplest cubic fields
Autoři
Gil-Muñoz, D.; Tinková, M.
Rok
2025
Publikováno
The Ramanujan Journal. 2025, 66(3), ISSN 1382-4090.
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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.
Additively indecomposable quadratic forms over totally real number fields
Autoři
Tinková, M.; Yatsyna, P.
Rok
2025
Publikováno
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES. 2025, 112(5), ISSN 0024-6107.
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We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper bounds for the minimal ranks of n-universal quadratic forms. For Q(sqrt 2),Q(sqrt 3),Q(sqrt 5),Q(sqrt 6), and Q(sqrt 21), we classify, up to equivalence, all classical, additively indecomposable binary quadratic forms.
Arithmetic of cubic number fields: Jacobi–Perron, Pythagoras, and indecomposables
Autoři
Kala, V.; Sgallová, E.; Tinková, M.
Rok
2025
Publikováno
Journal of Number Theory. 2025, 273 37-95. ISSN 0022-314X.
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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.
Bounds on the Pythagoras number and indecomposables in biquadratic fields
Autoři
Rok
2025
Publikováno
Proceedings of the Edinburgh Mathematical Society. 2025, 68(3), 843-868. ISSN 0013-0915.
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We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least 6. We also provide an upper bound on the norm and the minimal (codifferent) trace of additively indecomposable integers in some families of these fields.
The lifting problem for universal quadratic forms over simplest cubic fields
Autoři
Gil-Muñoz, D.; Tinková, M.
Rok
2024
Publikováno
Bulletin of the Australian Mathematical Society. 2024, 110(1), 77-89. ISSN 0004-9727.
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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
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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
Autoři
Rok
2023
Publikováno
The Ramanujan Journal. 2023, 61(4), 1121-1144. ISSN 1382-4090.
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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.