In recent years, the combinatorial properties of monomials ideals [7, 10, 14] and binomial ideals [9, 10] have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. The first is S[Λ]-resolutions of Λ-invariant submodules of k[Z] where Λ i...

The ring Z consists of the integers of the field Q, and Dedekind takes the theory of unique factorization in Z to be clear and well understood. The problem is that unique factorization can fail when one considers the integers in a finite extension of the rationals, Q(α). Kummer showed that when Q(α) is a cyclotomic extension (i.e. α is a primitive pth root of unity for a prime number p), one ca...

This paper, which expands on a talk given at the International Workshop on Fat Points, February 9-12, 2000, in Naples, Italy, surveys problems and progress on certain problems involving numerical characters for ideals I(Z) defining fat points subschemes Z = m1p1 + · · ·+ mnpn ⊂ P 2 for general points pi. In addition to presenting some new results, a collection of MACAULAY 2 scripts for computin...

In this paper, I present a new decision procedure for the ideal membership problem for polynomial rings over principal domains using discrete valuation domains. As a particular case, I solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker’s problem”, that is the problem of finding a decision procedure for the ideal membership prob...

We show that the thermodynamics of ideal gases may be derived solely from the Democritean concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion, aside from the law of energy conservation. Only a single corpuscle in contact with a heat bath submitted to a z and t-invariant force is considered. Most of the end results a...

Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a “connector” which is a linear combination of circuits with a square-free term. In particular if each circuit of IA with non-square-free terms is balanced, then IA is generated by circuits. As a consequence we prove that the toric id...

Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can impose on it to necessitate analogs of unique factorization. The trivial ideal structure of a field, the extent to which primary decomposition is unique, that a Noetherian ring necessarily has one, that a principal ideal domain is a unique factorization domain, and that a Dedekind domain h...

Let R be a 2-torsion free σ-prime ring. It is shown here that if U 6⊂ Z(R) is a σ-Lie ideal of R and a, b in R such that aUb = σ(a)Ub = 0, then either a = 0 or b = 0. This result is then applied to study the relationship between the structure of R and certain automorphisms on R. To end this paper, we describe additive maps d : R −→ R such that d(u) = 2ud(u) where u ∈ U, a nonzero σ-square close...

In this paper, we extend the characterization of Z[x]/〈f〉, where f ∈ Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay-Buchberger Basis Theorem) to rings, i.e. A[x1, . . . , xn]/a, wher...

X iv :m at h/ 01 10 09 7v 1 [ m at h. A G ] 9 O ct 2 00 1 LOCAL COMPLETE INTERSECTIONS IN P AND KOSZUL SYZYGIES DAVID COX AND HAL SCHENCK Abstract. We study the syzygies of a codimension two ideal I = 〈f1, f2, f3〉 ⊆ k[x, y, z]. Our main result is that the module of syzygies vanishing (schemetheoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local complete i...