Related papers: Binomial Difference Ideal and Toric Difference Var…

Lazard and Rouillier in [9], by introducing the concept of discriminant variety, have described a new and efficient algorithm for solving parametric polynomial systems. In this paper we modify this algorithm, and we show that with our…

Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X)+1 homogeneous polynomials that don't…

This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose…

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of…

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's…

With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems ($ABCD$) with respect to the integer and root lattices. As a result, we obtain a full description of…

Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$…

In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The…

We describe explicitly all multisets of weights whose defining projective toric varieties are self-dual. In addition, we describe a remarkable and unexpected combinatorial behaviour of the defining ideals of these varieties. The toric ideal…

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum…

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient…

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales…

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Groebner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show…

In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $\bz^n\oplus T$ with no invertible elements, where $T$ is a finite abelian…

A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…

Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give…

We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that…

A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of…

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can…

We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we…