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Mutations of fake weighted projective spaces (2014)
Journal Article
Coates, T., Gonshaw, S., Kasprzyk, A. M., & Nabijou, N. (2014). Mutations of fake weighted projective spaces. Electronic Journal of Combinatorics, 21(4), Article P4.14

We characterise mutations between fake weighted projective spaces, and give explicit formulas for how the weights and multiplicity change under mutation. In particular, we prove that multiplicity-preserving mutations between fake weighted projective... Read More about Mutations of fake weighted projective spaces.

Seven new champion linear codes (2013)
Journal Article
Brown, G., & Kasprzyk, A. M. (2013). Seven new champion linear codes. LMS Journal of Computation and Mathematics, 16, https://doi.org/10.1112/S1461157013000041

We exhibit seven linear codes exceeding the current best known minimum distance d for their dimension k and block length n. Each code is defined over F?, and their invariants [n,k,d] are given by [49,13,27], [49,14,26], [49,16,24], [49,17,23], [49,19... Read More about Seven new champion linear codes.

Small polygons and toric codes (2013)
Journal Article
Brown, G., & Kasprzyk, A. M. (2013). Small polygons and toric codes. Journal of Symbolic Computation, 51, https://doi.org/10.1016/j.jsc.2012.07.001

We describe two different approaches to making systematic classifications of plane lattice polygons, and recover the toric codes they generate, over small fields, where these match or exceed the best known minimum distance. This includes a [36,19,12]... Read More about Small polygons and toric codes.

Mirror symmetry and Fano manifolds (2013)
Conference Proceeding
Coates, T., Corti, A., Galkin, S., Golyshev, V., & Kasprzyk, A. M. (2013). Mirror symmetry and Fano manifolds.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

Fano polytopes (2012)
Book Chapter
Kasprzyk, A. M., & Nill, B. (2012). Fano polytopes. In A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, & E. Scheidegger (Eds.), Strings, gauge fields, and the geometry behind: the legacy of Maximilian Kreuzer (349-364). World Scientific. https://doi.org/10.1142/9789814412551_0017

Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for different classes of Fano polytopes.

Reflexive polytopes of higher index and the number 12 (2012)
Journal Article
Kasprzyk, A. M., & Nill, B. (2012). Reflexive polytopes of higher index and the number 12. Electronic Journal of Combinatorics, 19(3), Article P9

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a ch... Read More about Reflexive polytopes of higher index and the number 12.

Minkowski polynomials and mutations (2012)
Journal Article
Akhtar, M., Coates, T., Galkin, S., & Kasprzyk, A. M. (2012). Minkowski polynomials and mutations. Symmetry, Integrability and Geometry: Methods and Applications, 8, Article 094, pp. 707. https://doi.org/10.3842/SIGMA.2012.094

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Lauren... Read More about Minkowski polynomials and mutations.

Roots of Ehrhart polynomials of smooth Fano polytopes (2011)
Journal Article
Hegedüs, G., & Kasprzyk, A. M. (2011). Roots of Ehrhart polynomials of smooth Fano polytopes. Discrete and Computational Geometry, 46(3), https://doi.org/10.1007/s00454-010-9275-y

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly.... Read More about Roots of Ehrhart polynomials of smooth Fano polytopes.

The boundary volume of a lattice polytope (2011)
Journal Article
Hegedüs, G., & Kasprzyk, A. M. (2011). The boundary volume of a lattice polytope. Bulletin of the Australian Mathematical Society, 85(1), https://doi.org/10.1017/S0004972711002577

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(?P) is derived in terms of the number of boundary lattice points on the first [d/2] dilations of P. As an application we give a necessary and sufficient condition fo... Read More about The boundary volume of a lattice polytope.

On the combinatorial classification of toric log del Pezzo surfaces (2010)
Journal Article
Kasprzyk, A. M., Kreuzer, M., & Nill, B. (2010). On the combinatorial classification of toric log del Pezzo surfaces. LMS Journal of Computation and Mathematics, 13, https://doi.org/10.1112/S1461157008000387

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in t... Read More about On the combinatorial classification of toric log del Pezzo surfaces.

A Note on Palindromic ? -Vectors for Certain Rational Polytopes (2008)
Journal Article
Fiset, M., & Kasprzyk, A. M. (2008). A Note on Palindromic ? -Vectors for Certain Rational Polytopes. Electronic Journal of Combinatorics, 15, Article N18. https://doi.org/10.37236/893

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart ?-vector of P is palindromic. Perhaps less well-known is that a similar resu... Read More about A Note on Palindromic ? -Vectors for Certain Rational Polytopes.

Toric Fano three-folds with terminal singularities (2006)
Journal Article
Kasprzyk, A. M. (2006). Toric Fano three-folds with terminal singularities. TÔhoku Mathematical Journal, 58(1), 101-121. https://doi.org/10.2748/tmj/1145390208

This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only... Read More about Toric Fano three-folds with terminal singularities.