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Vítězslav Kala

Publications and source records attributed to Vítězslav Kala.

31 records · Page 2Linked to original sources
arXiv2019

Idempotence of finitely generated commutative semifields

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

math.AC↗
arXiv2019

Additive structure of totally positive quadratic integers

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.

math.NT↗
arXiv2017

On the rank of universal quadratic forms over real quadratic fields

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 coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.

math.NT↗
arXiv2017

Periodic representations in algebraic bases

We study periodic representations in number systems with an algebraic base $β$ (not a rational integer). We show that if $β$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(β)$ admits an eventually periodic representation with base $β$ and digits in $\mathcal A$.

math.NT↗
arXiv2017

Universal quadratic forms over multiquadratic fields

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.

math.NT↗
arXiv2016

Fermat's Last Theorem and Catalan's Conjecture in Weak Exponential Arithmetics

We study Fermat's Last Theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions (B,e) of models of arithmetical theories (in the language L=(0,1,+,x,<)) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms Exp. We construct a model (B,e) of Th(N) + Exp and a substructure (A,e) with e total and A model of Pr (Presburger arithmetic) such that in both (B,e) and (A,e) Fermat's Last Theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples a,b,c. On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in Th(N)+Exp (even in a weaker theory) and thus holds in (B,e) and (A,e). Finally, we also show that Fermat's Last Theorem for e is provable (again, under the assumption of ABC in N) in Th(N)+Exp+"coprimality for e".

math.LO↗
arXiv2016

Distribution of class numbers in continued fraction families of real quadratic fields

We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2, \dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ and generalize well-known families such as Chowla's $4n^2+1$, for which analogous results were recently proved by Dahl and Lamzouri.

math.NT↗
arXiv2016

Semifields and a theorem of Abhyankar

Abhyankar proved that every field of finite transcendence degree over $\mathbb{Q}$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $\mathbb{Z}[T_1, \dots, T_n]$ (for some $n$ depending on the field). We conjecture that his result can not be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

math.AC↗
arXiv2016

Norms of indecomposable integers in real quadratic fields

We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt D$ has the periodic continued fraction expansion $[u_0, u_1, u_2, \dots, u_{s-1}, 2u_0, u_1, u_2, \dots]$. This enables us to disprove a conjecture of Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable integer.

math.NT↗
arXiv2015

Lattice-ordered abelian groups finitely generated as semirings

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing that each such $\ell$-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici [8]. Then we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell$-groups associated to rooted trees (Theorem 4.1). This classification result has a number of important applications: for example it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot)$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture 1.1 discussed in [2], [15].

math.GR↗
arXiv2013

Congruences for Ramanujan's f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.

math.NT↗