Computation of Minimal Homogeneous Generating Sets and Minimal Standard Bases for Ideals of Free Algebras

Let \KX =K\langle X_1,\ldots ,X_n\rangle be the free algebra generated by $X=\{ X_1,\ldots ,X_n\}$ over a field $K$. It is shown that with respect to any weighted $\mathbb{N}$-gradation attached to \KX, minimal homogeneous generating sets for finitely generated graded (two-sided) ideals of \KX can be algorithmically computed, and that if an ungraded (two-sided) ideal $I$ of \KX has a finite Gr\"obner basis \G with respect to a graded monomial ordering on \KX, then a minimal standard basis for $I$ can be computed via computing a minimal homogeneous generating set of the associated graded ideal \langle\LH (I)\rangle.Comment: 13 pages. Algorithm1, Algorithm 2, and Algorithm 3 are revise

A Constructive Characterization of Solvable Polynomial Algebras

For the solvable polynomial algebras introduced and studied by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990)], a constructive characterization is given in terms of Gr\"obner bases for ideals of free algebras, thereby solvable polynomial algebras are completely determinable in a computational way.Comment: 12 pages with minor changes in Theorem 2.5 and its proo

On the Construction of Gr\"obner Bases with Coefficients in Quotient Rings

Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in $\Lambda$. It is shown that linear equations are solvable in $R$ (thereby theoretically Gr\"obner bases for ideals of $A$ are well defined and constructible) and that practically Gr\"obner bases in $A$ with respect to any given monomial ordering can be obtained by constructing Gr\"obner bases in $T$, and moreover, all basic applications of a Gr\"obner basis at the level of $A$ can be realized by a Gr\"obner basis at the level of $T$. Typical applications of this result are demonstrated respectively in the cases where $\Lambda=D$ is a PID, $\Lambda =D[y_1,...,y_m]$ is a polynomial ring over a PID $D$, and $\Lambda =K[y_1,...,y_m]$ is a polynomial ring over a field $K$.Comment: 21page