22 citations · 123 across the 24 of their papers we have counts for
24 papers
Toricness and smoothness criteria for spherical varieties
Giuliano Gagliardi, Johannes Hofscheier, Heath Pearson
We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orb…
Is there a smooth lattice polytope which does not have the integer decomposition property?
Johannes Hofscheier, Alexander Kasprzyk
We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question…
Examples of IDP lattice polytopes with non-log-concave -vector
Johannes Hofscheier, Vadym Kurylenko, Benjamin Nill
Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a th dilation is a sum of lattice points in the polyto…
The generalised Mukai conjecture for spherical varieties
Giuliano Gagliardi, Johannes Hofscheier, Heath Pearson
We prove the generalised Mukai conjecture for -factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absol…
Classification and Ehrhart Theory of Denominator 2 Polygons
Girtrude Hamm, Johannes Hofscheier, Alexander Kasprzyk
We present an algorithm for growing the denominator polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small …
Spherical amoebae and a spherical logarithm map
Victor Batyrev, Megumi Harada, Johannes Hofscheier +1
Let be a connected reductive algebraic group over with a maximal compact subgroup . Let be a (quasi-affine) spherical homogeneous space. In the first part…
Machine Learning the Dimension of a Polytope
Tom Coates, Johannes Hofscheier, Alexander Kasprzyk
We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machi…
Generalised Flatness Constants: A Framework Applied in Dimension
Giulia Codenotti, Thomas Hall, Johannes Hofscheier
Let and be a bounded set. Affine transformations given by an automorphism of and a translation in …
Polytopes and Machine Learning
Jiakang Bao, Yang-Hui He, Edward Hirst +3
We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflex…
Hilbert Series, Machine Learning, and Applications to Physics
Jiakang Bao, Yang-Hui He, Edward Hirst +3
We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\s…
Cohomology rings of toric bundles and the ring of conditions
Johannes Hofscheier, Askold Khovanskii, Leonid Monin
The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes.Puk…
Generalized flatness constants, spanning lattice polytopes, and the Gromov width
Gennadiy Averkov, Johannes Hofscheier, Benjamin Nill
In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular…
Splittings of Toric Ideals
Giuseppe Favacchio, Johannes Hofscheier, Graham Keiper +1
Let be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal can be "split" into the sum of two smaller toric…
Betti numbers of toric ideals of graphs: A case study
Federico Galetto, Johannes Hofscheier, Graham Keiper +3
We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this famil…
Containment Relations among Spherical Subgroups
Johannes Hofscheier
A closed subgroup of a connected reductive group is called if a Borel subgroup in has an open orbit on . We give a combinatorial characterizat…
Smooth centrally symmetric polytopes in dimension 3 are IDP
Matthias Beck, Christian Haase, Akihiro Higashitani +4
In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric -dimensional polytopes, by s…
Spanning Lattice Polytopes and the Uniform Position Principle
Johannes Hofscheier, Lukas Katthän, Benjamin Nill
A lattice polytope is called IDP if any lattice point in its th dilate is a sum of lattice points in . In 1991 Stanley proved a strong inequality in Ehrhart theory fo…
On Ehrhart polynomials of lattice triangles
Johannes Hofscheier, Benjamin Nill, Dennis Öberg
The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P),i(P)) where b(P) equals the number of lattice points on the boundary and i(P) equals the nu…
Ehrhart Theory of Spanning Lattice Polytopes
Johannes Hofscheier, Lukas Katthän, Benjamin Nill
A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes…
The generalized Mukai conjecture for symmetric varieties
Giuliano Gagliardi, Johannes Hofscheier
We associate to any complete spherical variety a certain nonnegative rational number , which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X -…
Homogeneous spherical data of orbits in spherical embeddings
Giuliano Gagliardi, Johannes Hofscheier
Let be a connected reductive complex algebraic group. Luna assigned to any spherical homogeneous space a combinatorial object called a homogeneous spherical datum. By a t…
Lattice polytopes, finite abelian subgroups in $\SL(n,\C)$ and coding theory
Victor Batyrev, Johannes Hofscheier
We consider -dimensional lattice polytopes with -polynomial for and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $…
Gorenstein spherical Fano varieties
Giuliano Gagliardi, Johannes Hofscheier
We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano vari…
A generalization of a theorem of G. K. White
Victor Batyrev, Johannes Hofscheier
An n-dimensional simplex Δ in \R^n is called empty lattice simplex if Δ\cap\Z^n is exactly the set of vertices of Δ. A theorem of G. K. White shows that if n=3 then any empty latti…