On Computing Janet Bases for Degree Compatible Orderings

Applications of Bar Code to involutive divisions and a greedy algorithm for complete sets

In this paper, we describe how to get Janet decomposition for a finite set of terms and detect completeness of that set by means of the associated Bar Code. Moreover, we explain an algorithm to find a variable ordering (if it exists) s.t. a given set of terms is complete according to that ordering. The algorithm is greedy and constructs a Bar Code from the maximal to the minimal variable, adjusting the variable ordering with a sort of backtracking technique, thus allowing to construct the desired ordering without trying all the n! possible orderings

On Exact Solvability of Anharmonic Oscillators in Large Dimensions

General Schrödinger equation is considered with a central polynomial potential depending on $2q$ arbitrary coupling constants. Its exceptional solutions of the so called Magyari type (i.e., exact bound states proportional to a polynomial of degree $N$) are sought. In any spatial dimension $D \geq 1$, this problem leads to the Magyari's system of coupled polynomial constraints, and only purely numerical solutions seem available at a generic choice of $q$ and $N$. Routinely, we solved the system by the construction of the Janet bases in a degree-reverse-lexicographical ordering, followed by thei…

Term-ordering free involutive bases

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 involut…

多项式理想的合冲基

Involutive Bases of Polynomial Ideals

本文提出一种基于合冲单项式除法的通用算法，用于构造多项式理想的合冲基，并证明其为Gröbner基的特殊冗余形式，同时引入优化策略减少计算量。

极小对合基

Minimal Involutive Bases

本文提出构造极小对合多项式基的算法，定义两种新对合除法并证明其性质，确保算法正确终止，且极小对合基在给定序下唯一，可作典范基。

不使用项序计算边界基

Computing Border Bases without using a Term Ordering

本文扩展了原始的 Border Basis Algorithm，使其能够计算不依赖于任何项序的边界基，突破了原算法需依赖度兼容项序的限制。