We propose a method for the approximation of the solution of high-dimensional problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of an ideal minimal residual method with a residual norm corresponding to the error in a solution norm of interest. We introduce and analyze an algorithm for the approximation of the best approximation i...

There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve C in affine n-space over an algebraically closed field K, provided C is not a local complete intersection. The existence of such an algorithm follows from the fact that given d ∈ N, there exists d ∈ N, such that if a is a height n−1 radical ideal inK[X1, . . . ,Xn], generated by polynomials of ...

Let A = {a1, . . . , am} ⊂ Z be a vector configuration and IA ⊂ K[x1, . . . , xm] its corresponding toric ideal. We completely determine the number of different minimal systems of binomial generators of IA. We also prove that generic toric ideals are generated by indispensable binomials. We associate to A a simplicial complex ∆ind(A). We show that the vertices of ∆ind(A) correspond to the indis...

We give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m×m minors of an m×n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This ...

We introduce a monomial ideal whose standard monomials encode the vertices of all fibers of a lattice. We study the minimal generators, the radical, the associated primes and the primary decomposition of this ideal, as well as its relation to initial ideals of lattice ideals.

Given a parametric polynomial ideal I, the algorithm DISPGB, introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Gröbner bases depending on the values of the parameters. An improvement using a discriminant ideal to rewrite the tree was described by Manubens and the author in 2005. In this paper we describe how to iterate the use ...

The core of an R-ideal I is the intersection of all reductions of I. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipman’s notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: core(I) is a ...

An old question of T. Jech and K. Prikry asks if an existence of a precipitous ideal implies necessary existence of a normal precipitous ideal. The aim of the paper is to prove some results in the positive direction. Thus, it is shown that under some mild assumptions, an existence of a precipitous ideal over א1 implies an existence of a normal precipitous ideal over א1 once a Cohen subset is ad...

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...