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We describe several analytical (i.e., precise) results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm. [ABSTRACT FROM AUTHOR]

We describe several analytical (i.e., precise) results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm. [ABSTRACT FROM AUTHOR]

We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibles for the monoid of multiplicities on an arbitrary matroid. [ABSTRACT FROM AUTHOR]

The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude. [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 describe several analytical results obtained in four candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes. [ABSTRACT FROM AUTHOR]

We give a simple proof for the fact that the Castelnuovo–Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals. [ABSTRACT FROM AUTHOR]

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very uniform primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen–Macaulay, Koszul and defined by a Gröbner basis of quadrics. [ABSTRACT FROM AUTHOR]

We establish a bound for the length of vectors involved in a unimodular triangulation of simplicial cones. The bound is exponential in the square of the logarithm of the multiplicity, and improves previous bounds significantly. The proof is based on a successive reduction of the highest prime divisor of the multiplicity and uses the prime number theorem to control the length of the subdividing vectors. [ABSTRACT FROM AUTHOR]

In this note, we characterize the Hilbert regularity of the Stanley-Reisner ring in terms of the -vector and the -vector of a simplicial complex . We also compute the Hilbert regularity of a Gorenstein algebra. [ABSTRACT FROM AUTHOR]

We introduce a partial order on the set of all normal polytopes in $$\mathbb R^d$$ . This poset $${{\mathsf {NPol}}}(d)$$ is a natural discrete counterpart of the continuum of convex compact sets in $$\mathbb R^d$$ , ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in $${{\mathsf {NPol}}}(d)$$ , called quantum jumps. The existence of extremal objects in $${{\mathsf {NPol}}}(d)$$ is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in $${{\mathsf {NPol}}}(5)$$ have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in $${{\mathsf {NPol}}}(4)$$ and $${{\mathsf {NPol}}}(5)$$ , found by a combination of the developed theory, random generation, and extensive computer search. [ABSTRACT FROM AUTHOR]

We describe the use of pyramid decomposition in Normaliz, a software tool for the computation of Hilbert bases and enumerative data of rational cones and affine monoids. Pyramid decomposition in connection with efficient parallelization and streamlined evaluation of simplicial cones has enabled Normaliz to process triangulations of size ≈ 5 ⋅ 10 11 that arise in the computation of Ehrhart series related to the theory of social choice. [ABSTRACT FROM AUTHOR]

We describe an algorithm for the computation of generalized (or weighted) Ehrhart series based on Stanley decompositions as implemented in the offspring NmzIntegrate of Normaliz. The algorithmic approach includes elementary proofs of the basic results. We illustrate the computations by examples from combinatorial voting theory. [ABSTRACT FROM AUTHOR]

An ideal I in a polynomial ring S has linear powers if all the powers Ik of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers. [ABSTRACT FROM AUTHOR]

It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations of -minors for . We will exhibit minimal relations in degrees (non-Plücker in general) and , and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group. [ABSTRACT FROM AUTHOR]

Abstract: Normaliz is a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems. In this paper we present the algorithms implemented in the program. [Copyright &y& Elsevier]

Abstract: Let V and W be vector spaces of dimension m and n respectively. We investigate the Zariski closure of the image of the map , . In the case , is the cone over a Grassmannian, but for one has . We analyze the -orbits in via the G-stable prime ideals in . It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in arise from the images for and simple algebraic operations. In the last section we determine the singular locus of . Apart from well-understood exceptional cases, it is formed by the elements of rank ⩽1 in . [Copyright &y& Elsevier]

Abstract: We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley–Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster''s decomposition of local cohomology of Stanley–Reisner rings. [Copyright &y& Elsevier]

Abstract: We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen''s g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, , can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies. [Copyright &y& Elsevier]

Abstract: Given a seminormal affine monoid M we consider several monoid properties of M and their connections to ring properties of the associated affine monoid ring over a field K. We characterize when satisfies Serre''s condition () and analyze the local cohomology of . As an application we present criteria which imply that is Cohen–Macaulay and we give lower bounds for the depth of . Finally, the seminormality of an arbitrary affine monoid M is studied with characteristic p methods. [Copyright &y& Elsevier]

Abstract: We compute the canonical class of certain Rees algebras. Our formula generalizes that of Herzog and Vasconcelos. Its proof relies on the fact that the formation of the canonical module commutes with subintersections in important cases. As an application we treat the classical determinantal ideals and the corresponding algebras of minors. [Copyright &y& Elsevier]

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]

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski''s theory of complete ideals and of representation theory. [Copyright &y& Elsevier]

Provides an overview of the developments of problems and algorithms for affine semigroups or discrete analogues of convex polyhedral cones. Affine semigroups and its properties; Algebraic properties for the corresponding rings; Link provided by the polytopal semigroups and algebras with the classical theme of triangulations into unimodular simplices.

Provides an overview on the `tight closure' theory, a synonym for `characteristic p methods in commutative algebra,' developed by Mel Hochster and Craig Huneke. Ability to tie up invariant theory, rational singularities, the Briancon-Skoda theorem, the `homological conjectures and Cohen-Macaulay modules and algebras.