Lifting Grobner bases from the exterior algebra

Efficient Gröbner Bases Computation over Principal Ideal Rings

In this paper we present a new efficient variant to compute strong Gröbner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gröbner basis computations to Gröbner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gröbner basis alg…

多项式理想的Gröbner基的存在性与构造

Existence and Construction of a Gröbner Basis for a Polynomial Ideal

该研究针对交换环R上多项式环的理想，提出一种Gröbner基算法的提升构造方法，前提是R自身每个理想都有Gröbner基，推进了多项式理想Gröbner基的存在性与构造研究。

Algebraic shifting and exterior and symmetric algebra methods

We establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shift is the best among the exterior shifting operations in the sense that it increases the number of facets the least. Methods of proof include Gröbner basis theory over the exterior algebra, Cartan homology, degree functions, and Alexander duality.

Lifting Markov Bases and Higher Codimension Toric Fiber Products

We study how to lift Markov bases and Gröbner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.

Equality of Graver bases and universal Gröbner bases of colored partition identities

Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gröbner basis and the Graver basis. While the former is generally a proper subset of the latter, there are cases for which the two sets coincide. The most prominent examples among them are toric ideals of unimodular matrices. Equality of universal Gröbner basis and Graver basis is a combinatorial property of the toric ideal (or, of the defining matrix), providing interesting information about ideals of higher Lawrence liftings of a mat…

Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in SINGULAR. Our modular algorithms to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verificat…