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Let $k$ be a field, $ mathcal{L}subset mathbb{Z}^n$ be a lattice such that $Lcap mathbb{N}^n={{bf 0}}$, and $I_Lsubset Bbbk[x_1,..., x_n]$ the corresponding lattice ideal. We present the generalized Scarf complex of $I_L$ and show that it is indispensable in the sense that it is contained in every minimal free resolution of $R/I_L$.
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $bz^noplus T$ with no invertible elements, where $T$ is a finite abelian group. We also ch
In this article we associate to every lattice ideal $I_{L,rho}subset K[x_1,..., x_m]$ a cone $sigma $ and a graph $G_{sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $sigma $. To every polynomial $F$ we assign a subgraph
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We pro
The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relati
Let $R=mathbf{C}[xi_1,xi_2,ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $mathfrak{S}$. We classify the $mathfrak{S}$-primes of $R$, determine the containments among these ideals, a