Publisher Summary This chapter discusses the monomial algebras and its connections to combinatorics, graph theory, and polyhedral geometry. Some important notions from commutative algebra that have played a role in the development of the theory, such as Cohen-Macaulay ring, normal ring, Gorenstein ring, integral closure, Hilbert series, and local cohomology are introduced. The upper bound theorem for the number of faces of a simplicial sphere, a description of the integral closure of an edge subring, a generalized marriage theorem for a certain family of graphs, and a study of systems of binomials in the ideal of an affine toric variety are provided as applications. It illustrates the interplay between several areas of mathematics and the power of combinatorial commutative algebra techniques. There is a connection between monomial rings and monomial subrings due to the fact that the initial ideal of a toric ideal is a monomial ideal. This allows computing several invariants of projective varieties using algebraic systems such as CoCoA and Macaulay2. An important tool to study monomial subrings is Normaliz, which is effective in practice and can be used to find normalizations, Hilbert series, Ehrhart rings, and volumes of lattice polytopes.

Monomial algebras and polyhedral geometry

Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields

Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

Monomial Algebras, Second Edition

Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.

The minimum distance of parameterized codes on projective tori

Geometry of toric varieties

A note on affine toric varieties

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.

Complete intersection vanishing ideals on degenerate tori over finite fields

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup. 
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Let k be an arbitrary field and C a toric set in the affine space A n given parametrically by monomials. Using linear algebra we give necessary and sufficient conditions for C to be an affine toric variety, and show some applications. © 2000 Elsevier Science Inc. All rights reserved.

Leticia Zárate

Papers

A note on affine toric varieties

Complete intersection vanishing ideals on degenerate tori over finite fields

Complete intersection vanishing ideals on degenerate tori over finite fields

A note on affine toric varieties