Volume 25, Issue 3

We study sets of univariate hyperbolic polynomials that share the same first few coefficients and show that they have a natural combinatorial description akin to that of polytopes. We define a stratification of these sets in terms of root arrangements of hyperbolic polynomials and show that any stratum is either empty, a point or of maximal dimension, and in the latter case we characterise its relative interior. This is used to show that the poset of strata is a graded, atomic and coatomic lattice and to provide an algorithm for computing which root arrangements are realised in these sets of hyperbolic polynomials.

In this note we give a short proof of Godbersen’s conjecture for the special class of locally anti-blocking bodies. We show that all equality cases amongst locally anti-blocking bodies are for locally anti-blocking simplices, further supporting the conjecture. The proof of the equality cases introduces a useful calculation of mixed volumes of aligned simplices.

Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or Bergman metric. We define another natural metric, from the generalized Hilbert metric defined in [4], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish–Chandra realization in projective spaces. We describe this construction for the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to the Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

For p an odd prime, let p0:=pf,q0:=p0t and q:=p02t−1, ${{p}_{0}}:={{p}^{f}},{{q}_{0}}:=p_{0}^{t}\text{ and }q:=p_{0}^{2t-1},$ where f and t are natural numbers and f is odd. The curve 𝒞 defined over the finite field Fqn ${{\mathbb{F}}_{{{q}^{n}}}}$ by the affine equation C : Y q − Y = X q 0 X q − X $$\mathcal{C}:{{Y}^{q}}-Y={{X}^{{{q}_{0}}}}\left( {{X}^{q}}-X \right)$$ is called the generalized Suzuki curve. For f = 1 it was studied extensively in [1]. In [19] the case f > 1 was addressed. It turns out that these curves constitute new examples of maximal curves for suitable values of p and n . Whereas in [19] cohomological methods and computations were used to determine the zeta functions of these curves, the aim of this article is to adapt the approach from [1] to the case f > 1.

Let ϕ be a collineation of order 3 acting on PG(2, q 3 ) whose fixed points are exactly an 𝔽 q -plane P 2, q . Let T be a point whose orbit under ϕ is a triangle and let S T be the subgroup of PGL(3, q 3 ) that fixes setwise the 𝔽 q -plane P 2, q and fixes setwise the line TϕTϕ2  ${{T}^{\phi }}{{T}^{{{\phi }^{2}}}}\,$ The point orbits of S T form a partition of the points of PG(2, q 3 ) and consist of: the singletons T,Tϕ,Tϕ2; $T,{{T}^{\phi }},{{T}^{{{\phi }^{2}}}};$ scattered linear sets on the sides of the triangle TTϕTϕ2; $T{{T}^{\phi }}{{T}^{{{\phi }^{2}}}};$ and 𝔽 q -planes. This article studies the structure of this partition, looking at maps that permute elements of the partition. The motivation in studying this partition lies in its application to the construction of the Figueroa projective planes, and the article concludes with a characterisation in this setting.

Recently, algebraic lattices have been widely considered for applications in coding theory and cryptography. Likewise, well-rounded lattices have proven useful for signal transmission in MIMO and SISO wiretap channels. Previous works have conducted extensive studies on well-rounded algebraic lattices. One of them demonstrates the existence of well-rounded lattices obtained through the Minkowski embedding, representing the image of ℤ-modules in the ring of integers of cyclic number fields 𝕂 of odd prime degree p , where p remains unramified in the extension 𝕂/ℚ. In this work, we consider the case where p is ramified in 𝕂/ℚ and introduce a family of ℤ-modules that realize well-rounded algebraic lattices in ℝ p for many (potentially infinitely many) values of p .

We give a description of the split Cayley hexagon, using the split Cayley algebra of octonions, and its subalgebras of dimension five or six as points or lines, respectively. That description is then used to identify various interesting subgeometries, and a local isomorphism from the hexagon onto a projective plane.

This paper is devoted to the study of the relative Lipschitz saturation of complex algebraic varieties. More precisely, we investigate the concept of Lipschitz saturation of a variety in another, and we focus on the case where the dominant morphism between the two varieties is not necessarily finite. In particular, we answer, in the case of algebraic varieties, an open question of Pham and Teissier concerning the finiteness of the Lipschitz saturation of general algebras. As an application, we provide algebraic criteria for two algebraic varieties to be linked by an algebraic morphism, which is a locally biLipschitz homeomorphism on the closed points of the variety.

This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Roldán-Pensado that the “prime grid” {( p , q ) ∈ ℤ 2 : p and q are prime} ⊆ ℝ 2 contains empty polygons with arbitrarily many vertices. This implies that no Helly-type theorem is true for the prime grid.

This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of Łojasiewicz’s inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.