arXiv subjects
Publications and source records attributed to Michela Ceria.
We present Buchberger Theory and Algorithm of Gröbner bases for multivariate Ore extensions of rings presented as modules over a principal ideal domain. The algorithms are based on Möller Lifting Theorem.
Aim of this paper is to count $0$-dimensional stable and strongly stable ideals in $2$ and $3$ variables, given their (constant) affine Hilbert polynomial. To do so, we define the Bar Code, a bidimensional structure representing any finite set of terms $M$ and allowing to desume many properties of the corresponding monomial ideal $I$, if $M$ is an order ideal. Then, we use it to give a connection between (strongly) stable monomial ideals and integer partitions, thus allowing to count them via known determinantal formulas.
We present here a new approach for computing Gröbner bases for bilateral modules over an effective ring. Our method is based on Weispfenning notion of restricted Gröbner bases and related multiplication.
In this paper, we consider a monomial ideal J in P := A[x1,...,xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals I in P such that the A-module P/I is free with basis given by the set of terms in the Groebner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Janet and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied for the special case of a strongly stable monomial ideal J. Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F(J), called stably complete, that allows an explicit description of the family Mf(J). We obtain stronger results if J is quasi stable, proving that F(J) is a Pommaret basis and Mf(J) has a natural structure of affine scheme. The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.
Given a finite set $\mathbf{X}$ of distinct points, Marinari-Mora's 'Axis of Evil Theorem' states that a combinatorial algorithm and interpolation enable to find a 'linear' factorization for a lexicographical minimal Groebner basis $\mathcal{G}(I(\mathbf{X}))$ of the zerodimensional radical ideal $I(\mathbf{X})$. In this work we provide such algorithm, showing that it ends in a finite number of steps and that it actually provides the correct result. The 'Axis of Evil' algorithm takes as input the monomial basis of the initial ideal $T(I(\mathbf{X}))$ but its starting point is the (finite) Groebner escalier $N$ (obtained via Cerlienco-Mureddu correspondence) so we will also define the `potential expansion' 's algorithm, a combinatorical algorithm which computes the minimal basis from a finite Groebner escalier.