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We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear-ordering polytopes whose Euclidean automorphism groups are determined. [ABSTRACT FROM AUTHOR]

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity m , and use it to prove Wilf's conjecture holds whenever m ≤ 1 8. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf's conjecture via a combinatorially flavored game played on the elements of a certain finite poset. [ABSTRACT FROM AUTHOR]

We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin''s theory and Quillen''s + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by polyhedral geometry. This paper is a continuation of Bruns and Gubeladze (Polyhedral K2, Manuscr. Math.) which is devoted to the study of polyhedral aspects of the classical Steinberg relations. The present work explores the polyhedral geometry behind Suslin''s well known proof of the coincidence of the classical Volodin''s and Quillen''s theories. We also determine all K-groups coming from two-dimensional polytopes. [Copyright &y& Elsevier]