Computing dimension and independent sets for polynomial ideals

Gröbner Bases Applied to Finitely Generated Field Extensions

“…If B is the set returned by the algorithm then by construction {Z i : x i ∈ B} is a maximal strongly independent subset of { Z} modulo P ( x)/k( g) , and since P ( x)/k( g) is prime any maximal subset of { Z} strongly independent modulo P ( x)/k( g) is also maximal independent modulo P ( x)/k( g) (this is conjectured in Kredel andWeispfenning, 1988 andproven in Kalkbrener andSturmfels, 1995). Hence with Becker and Weispfenning (1993, Lemma 7.25) we may conclude that the residue classes of the Z i where…”

On Gröbner bases and Krull dimension of residue class rings of polynomial rings over integral domains

“…We extend the concept of strongly independent indeterminates modulo a introduced in (Kredel & Weispfenning, 1988) for ideals in k[x 1 , . .…”

“…Algorithm 3 also determines the maximal independent set of indeterminates modulo a. This algorithm is along the lines of the algorithm described in (Kredel & Weispfenning, 1988, Section 3). …”

“…. , x n ]/a (Mora & Möller, 1983;Kredel & Weispfenning, 1988), thus providing an algorithmic framework for determining the Krull dimension of the affine variety associated with a. This paper studies the question of whether one can give Gröbner basis methods to compute the Krull dimension of A[x 1 , .…”

The computation of Gröbner bases on a shared memory multiprocessor

“…They are based on the formula above. Some other algorithms to compute the dimension of the ideal I once a Grobner basis has been computed have been designed, for example by Kredel and Weispfenning, [23]. The problem with all these is that the computation of a Grobner basis takes a long time, both in terms of worst complexity and of "observed" complexity.…”