Algebras Defined by Monic Gröbner Bases over Rings

Averaging algebras, rewriting systems and Gr$ö$bner-Shirshov bases

In this paper, we study the averaging operator by assigning a rewriting system to it. We obtain some basic results on the kind of rewriting system we used. In particular, we obtain a sufficient and necessary condition for the confluence. We supply the relationship between rewriting systems and Grobner-Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a non-empty set.

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{ö}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them ca…

On the generalized Clifford algebra of a monic polynomial

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial $Φ(Z,X_1,\dots,X_n)=Z^d-\sum_{k=1}^d f_k(X_1,\dots,X_n) Z^{d-k}$ of degree $d$ in $n+1$ variables over some field $F$. We completely determine its structure in the following cases: $n=2$ and $d=3$ and either $\operatorname{char}(F)=3$, $f_1=0$ and $f_2(X_1,X_2)=e X_1 X_2$ for some $e \in F$, or $\operatorname{char}(F) \neq 3$, $f_1(X_1,X_2)=r X_2$ and $f_2(X_1,X_2)=e X_1 X_2+t X_2^2$ for some $r,t,e \in F$. Except for a few exceptions, this algebra…

Valuation Extensions of Algebras Defined by Monic Gröbner Bases

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrowΓ\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free $K$-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on $X_1,...,X_n$. If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gröbner basis of ${\cal I}$ in $K\langle X\rangle$, where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$-algebra on $X_1,...,X_n$, t…

凯莱 - 迪克森代数上的一般多项式与特征值

General Polynomials and Eigenvalues Over Cayley--Dickson Algebras

研究凯莱 - 迪克森代数上首一二次一般多项式零集，给出求根方法，证明实数上局部复凯莱 - 迪克森代数中此类多项式必有根，以及凯莱八元数上2×2矩阵右特征值存在并可计算部分右特征值。

Hopf范畴的自由与余自由构造

Free and co-free constructions for Hopf categories

本文证明在温和条件下Hopf范畴的范畴是局部可表示的，推导出自由与余自由Hopf范畴的存在性，给出半Hopf范畴上的显式构造，并引入“非常平坦幺半积”概念在模范畴中进行研究。