MATHEMATICA SCANDINAVICA Mathematica Scandinavica en-US MATHEMATICA SCANDINAVICA 0025-5521 A dynamical analogue of a question of Fermat <p>Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.</p> Mohammad Sadek Tugba Yesin Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-142342 Invertible objects in Franke's comodule categories <p>We study the Picard group of Franke's category of quasi-periodic $E_0E$-comodules for $E$ a 2-periodic Landweber exact cohomology theory of height $n$ such as Morava $E$-theory, showing that for $2p-2 &gt; n^2+n$, this group is infinite cyclic, generated by the suspension of the unit. This is analogous to, but independent of, the corresponding calculations by Hovey and Sadofsky in the $E$-local stable homotopy category. We also give a computation of the Picard group of $I_n$-complete quasi-periodic $E_0E$-comodules when $E$ is Morava $E$-theory, as studied by Barthel-Schlank-Stapleton for $2p-2 \ge n^2$ and $p-1 \nmid n$, and compare this to the Picard group of the $K(n)$-local stable homotopy category, showing that they agree up to extension.</p> Drew Heard Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-142361 Voronoi cells in metric algebraic geometry of plane curves <p>Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.</p> Madeline Brandt Madeleine Weinstein Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-139875 Gauge-invariant uniqueness theorems for $P$-graphs <p>We prove a version of the result in the title that makes use of maximal coactions in the context of discrete groups. Earlier Gauge-Invariant Uniqueness theorems for $C^*$-algebras associated to $P$-graphs and similar $C^*$-algebras exploited a property of coactions known as normality. In the present paper, the view point is that maximal coactions provide a more natural starting point to state and prove such uniqueness theorems. A byproduct of our approach consists of an abstract characterization of co-universal representations for a Fell bundle over a discrete group.</p> Robert Huben S. Kaliszewski Nadia S. Larsen John Quigg Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-139786 Splittings for $C^*$-correspondences and strong shift equivalence <p>We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to $C^*$-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of $C^*$-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular $C^*$-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant $*$-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for $C^*$-correspondences.</p> Kevin Aguyar Brix Alexander Mundey Adam Rennie Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-142308 Minimal piecewise linear cones in $\mathbb{R}^4$ <p>We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimal with respect to Lipschitz maps in the sense of [Almgren, F., Mem. Amer. Math. Soc. 4 (1976), no. 165] as in [Taylor, J. E., Ann. of Math. (2) 103 (1976), no. 3, 489–539]. There are three that arise naturally by taking products of $\mathbb{R}$ with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no piecewise linear minimizers outside these five.</p> Asgeir Valfells Copyright (c) 2024 MATHEMATICA SCANDINAVICA 2024-02-26 2024-02-26 130 1 10.7146/math.scand.a-140336