arXiv subjects
Publications and source records attributed to Lorenz Halbeisen.
We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-$ω_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\textsf{CH}$, there is a unique $Q$-point up to isomorphism. In particular, it is consistent that there is only one $Q$-point while there are $2^{\mathfrak{c}}$-many near-coherence classes of ultrafilters.
We consider the problem of finding integer triangles with $R/r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that for general triangles $R/r>1/4$ applies. The equation $R/r=N$ turns out to be related to the elliptic curve $\mathcal{E}_N$ given by $v^2=u^3+2(2N^2+2N-1)u^2-(4N-1)u$. If $N>1/4$ is rational, then the torsion group of $\mathcal{E}_N$ is $\mathbb Z/2\mathbb Z\times\mathbb Z/6\mathbb Z$ if $N(N+2)$ is a square and $\mathbb Z/6\mathbb Z$ otherwise. We show that a rational triangle with rational ratio $R/r=N$ exists if and only if $N>1/4$ and there exists a rational non-torsion point on the curve $\mathcal{E}_N$ which satisfies a certain condition. Furthermore, we show that the rank of $\mathcal{E}_N$ is positive when $N = m^2 \pm 1>1/4$ for a rational $m$. We also show that on every curve $\mathcal{E}_N$ whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with $R/r=N$.
We generalize the main result of arXiv:2505.17960 and show the consistency of the statement ``There are exactly $n$ $Q$-points up to isomorphism" for any finite $n$. Furthermore, we show that the above statement for $n=2$ can alternatively be obtained by a length-$ω_2$ countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.
A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$, such that $(a^hk, b^hl)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $Γ_{a^h ,b^h}$ for $h\ge 3$ with torsion group isomorphic to $\mathbb Z/2\mathbb Z \times \mathbb Z/4\mathbb Z$ such that $Γ_{a^h,b^h}$ has positive rank over $\mathbb Q$ if and only if $(a,b)$ is a pythapotent pair of degree $h$. As a side result, we get that if $(a, b)$ is a pythapotent pair of degree $h$, then there exist infinitely many pythagorean pairs $(k,l)$, not multiples of each other, such that $(a^hk,b^hl)$ is a pythagorean pair. In particular, we show that any pythagorean pair is always a pythapotent pair of degree 3. In a previous work, pythapotent pairs of degrees 1 and 2 have been studied.
The Hesse curve or Hesse derivative Hess$(Γ_f)$ of a cubic curve $Γ_{f}$ given by a homogeneous polynomial $f$ is the set of points $P$ such that $\det \left(H_f (P)\right)=0$, where $H_f (P)$ is the Hesse matrix of $f$ evaluated at $P$. Also Hess$(Γ_f)$ is again a cubic curve. We show that for a point $P\in$Hess$(Γ_{f})$, all the contact points of tangents from $P$ to the curves $Γ_{f}$ and Hess$(Γ_{f})$ are intersection points of two straight lines $\ell_1^P$ and $\ell_2^P$ (meeting on Hess$(Γ_{f})$) with $Γ_{f}$ and Hess$(Γ_{f})$, where the product of $\ell_1^P$ and $\ell_2^P$ is the polar conic of $Γ_{f}$ at $P$. The operator Hess defines an iterative discrete dynamical system on the set of the cubic curves. We identify the two fixed points of this system, investigate orbits that end in the fixed points, and discuss the closed orbits of the dynamical system.
The Ramsey Choice principle for families of $n$-element sets, denoted $\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$. We investigate for which positive integers $m$ and $n$ the implication $\mathrm{RC}_m \Rightarrow \mathrm{RC}_n$ is provable in ZF. It will turn out that beside the trivial implications $\mathrm{RC}_m \Rightarrow \mathrm{RC}_m$, under the assumption that every odd integer $n>5$ is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in ZF is $\mathrm{RC}_2 \Rightarrow \mathrm{RC}_4$.
For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF.
Let $P$ be an arbitrary point on an elliptic curve over the complex numbers of the form $y^2=x^3+a_4\,x+a_6$ or of the form $y^2=x^3+a_2\,x^2+a_4\,x$. We provide explicit formulae to compute the points $P/2$, i.e., the points $Q$ such that $2*Q=P$.
We give new parametrisations of elliptic curves in Weierstrass normal form $y^2=x^3+ax^2+bx$ with torsion groups $\mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ over $\mathbb{Q}$, and with $\mathbb{Z}/14\mathbb{Z}$ and $\mathbb{Z}/16\mathbb{Z}$ over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group $\mathbb{Z}/12\mathbb{Z}$ and positive rank. Furthermore, we found elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ and rank $3$, which is a new record for such curves, as well as some new elliptic curves with torsion group $\mathbb{Z}/16\mathbb{Z}$ and rank $3$.
We give a projective proof of the butterfly porism for cyclic quadrilaterals and present a general reversion porism for polygons with an arbitrary number of vertices on a conic. We also investigate projective properties of the porisms.
In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions.
A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2 = \Box$ (i.e., $a^2 + b^2$ is a square). A pythagorean pair $(a, b)$ is called a double-pythapotent pair if there is another pythagorean pair $(k,l)$ such that $(ak,bl)$ is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair $(k,l)$ which is not a multiple of $(a,b)$, such that $(a^2k,b^2l)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $Γ_{a,b}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/4\mathbb Z$, such that $Γ_{a,b}$ has positive rank if and only if $(a, b)$ is a double-pythapotent pair. Similarly, to each pythagorean pair $(a, b)$ we assign an elliptic curve $Γ_{a^2 ,b^2}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$, such that $Γ_{a^2,b^2}$ has positive rank if and only if $(a,b)$ is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve $Γ$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$ is isomorphic to a curve of the form $Γ_{a^2 ,b^2}$ , where $(a,b)$ is a pythagorean pair. As a side-result we get that if $(a,b)$ is a double-pythapotent pair, then there are infinitely many pythagorean pairs $(k, l)$, not multiples of each other, such that $(ak, bl)$ is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs.
Let $n\inω$. The weak choice principle $\operatorname{RC}_n$ states that for every infinite set $x$ there is an infinite subset $y\subseteq x$ with a choice function on $[y]^n:=\{z\subseteq y\mid \lvert z\rvert =n\}$. $\operatorname{C}_n^-$ states that for every infinite family of $n$-element sets, there is an infinite subfamily $\mathcal{G}\subseteq\mathcal{F}$ with a choice function. $\operatorname{LOC}_n^-$ and $\operatorname{WOC}_n^-$ are the same statement but we assume that the family $\mathcal{F}$ is linearly orderable ($\operatorname{LOC}_n^-$) or well-orderable ($\operatorname{WOC}_n^-$). In the first part of this paper we will give a full characterization of when the implication $\operatorname{RC}_m\Rightarrow \operatorname{WOC}_n^-$ with $m,n\inω$ holds in $\operatorname{ZF}$. We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part of we will show some generalizations. In particular we will show that $\operatorname{RC}_5\Rightarrow \operatorname{LOC}_5^-$ and $\operatorname{RC}_6\Rightarrow \operatorname{C}_3^-$, answering two open questions from Halbeisen and Tachtsis. Furthermore we will show that $\operatorname{RC}_6\Rightarrow \operatorname{C}_9^-$ and $\operatorname{RC}_7\Rightarrow \operatorname{LOC}_7^-$.
For every natural number $n$ we introduce a new weak choice principle $\mathrm{nRC_{fin}}$: Given any infinite set $x$, there is an infinite subset $y\subseteq x$ and a selection function $f$ that chooses an $n$-element subset from every finite $z\subseteq y$ containing at least $n$ elements. By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of $\mathrm{nRC_{fin}}$ to the weak choice principles $\mathrm{RC_m}$ (that has already been studied by Montenegro, Halbeisen and Tachtsis): Given any infinite set $x$, there is an infinite subset $y\subseteq x$ with a choice function $f$ on the family of all $m$-element subsets of $y$. Moreover, we prove a stronger analogue of Montenegros results when we study the relation between $\mathrm{nRC_{fin}}$ and $\mathrm{kC_{fin}^-}$ which is defined by: Given any infinite family $\mathcal{F}$ of finite sets of cardinality greater than $k$, there is an infinite subfamily $\mathcal{A}\subseteq \mathcal{F}$ with a selection function $f$ that chooses a $k$-element subset from each $A\in\mathcal{A}$.
We give an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of $m^2=n^2+nl+l^2$.
A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.
A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (e'_i) of a normalized basic sequence (x_i) in a Banach space X. We show how to generalize the construction to define a new creature (e_i), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem--a strengthened form of Ramsey's Theorem--to generate asymptotic models with a stronger form of convergence.
We define the dualizations of objects and concepts which are essential for investigating the Ramsey property in the first levels of the projective hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and dual Laver forcing, and show that the Harrington-Kechris techniques for proving the Ramsey property from determinacy work in the dualized case as well.