Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free resolution of S/M over S. The difficulty of this problem is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence [BH,Ho,St]) int...

These are lecture notes, in progress, on monomial ideals. The point of view is that monomial ideals are best understood by drawing them and looking at their corners, and that a combinatorial duality satisfied by these corners, Alexander duality, is key to understanding the more algebraic duality theories at play in algebraic geometry and commutative algebra. Sections written so far cover Alexan...

(June 15, 2016.) FIX: June 15, 2016. This needs to be re-written. Monomial transformations of linear codes are linear isometries for the Hamming weight. A code alphabet has the extension property for the Hamming weight when every linear isometry between codes extends to a monomial transformation. MacWilliams proved that finite fields have the extension property for the Hamming weight. In contra...

Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of the Rees algebra of I in terms of an Ehrhart ring. We introduce the basis Rees cone of a matroid (or a polymatroid) and study their facets. Some application...

Abstract. The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milne’s volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squaref...

Abstract. As a generalization of cyclic codes of length p over Fpa , we study n-dimensional cyclic codes of length p1 ×· · ·×pn over Fpa generated by a single “monomial”. Namely, we study multi-variable cyclic codes of the form 〈(x1 − 1) i1 · · · (xn − 1) n〉 ⊂ Fq [x1,...,xn] 〈x ps1 1 −1,...,x psn n −1〉 . We call such codes monomial-like codes. We show that these codes arise from the product of ...

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal I in a polynomial ring a monomial ideal in≺I, in some special situations the monomial ideal in≺I is square free. On the other hand given any monomial ideal I of a polynomial ring S, we can define the toric K[I] ⊂ S. In this paper we will study toric rings defined by Segre embed...

A monomial dynamical system f : K → K over a finite field K is a nonlinear deterministic time discrete dynamical system with the property that each component function fi : K n → K is a monic nonzero monomial function. In this paper we provide an algebraic and graph theoretic framework to study the dynamic properties of monomial dynamical systems over a finite field. Within this framework, chara...

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the m...