134 research outputs found
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
over a field . It is shown that with respect to any
weighted -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 of
\KX has a finite Gr\"obner basis \G with respect to a graded monomial
ordering on \KX, then a minimal standard basis for 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 be a commutative Noetherian ring, and let be a proper ideal
of , . Consider the polynomial rings and . Suppose that linear equations are
solvable in . It is shown that linear equations are solvable in
(thereby theoretically Gr\"obner bases for ideals of are well defined and
constructible) and that practically Gr\"obner bases in with respect to any
given monomial ordering can be obtained by constructing Gr\"obner bases in ,
and moreover, all basic applications of a Gr\"obner basis at the level of
can be realized by a Gr\"obner basis at the level of . Typical applications
of this result are demonstrated respectively in the cases where is
a PID, is a polynomial ring over a PID , and
is a polynomial ring over a field .Comment: 21page
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