A theoretical basis for the reduction of polynomials to canonical forms

Abstract

We define a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of polynomial ideals (e.g. the problem of constructing canonical forms for polynomials) is reducible to the construction of bases of this type. We prove a characterization theorem for these bases which immediately leads to an effective method for their construction.

Symbolic-Numerical Modeling of the Influence of Damping Moments on Satellite Dynamics

“…To find solutions of the algebraic system (2)- (3) we used the algorithm for constructing the Gröb-ner bases [4], implemented as the package Groebner[Basis] in the computer algebra system Maple 15 [5]. We constructed the Gröbner basis of the system of six second-order polynomials (2)-(3) with six variables a i j (i = 2, 3; j = 1, 2, 3), with respect to the lexicographic ordering of variables by using the option plex.…”

“…The polynomial from the Gröbner basis that depends on the variable a 33 is: a 33 (a 4 33 − 9a 2 33 + k 2 2 )((a 4 33 − 9a 2 33 + k 2 1 ) = 0. To determine all equilibria, it is required to consider separately the following three cases: a 22 = 1, a 22 = −1, and a 22 = 0.…”

The computation of Gröbner bases on a shared memory multiprocessor

“…(3) If there exist power products T 3 ,T 4j T 5 such that T x = T 3 T 4 , T 2 = T 3 T 5 and T 4 >^ T 5 , our induction hypothesis is that there exist k £ K and P £ A such that T 4 -k T 5 + P and T 5 is not a power product of P. Then, 7\ -kT 2 + T 3 P and T 2 is not a power product of T 3 P. • Proo/ of 2.2: If >^ is a transitive relation, in order not to be a strict order relation, it has to be either not antireflexive or not antisymmetric. By transitivity, if it is not antisymmetric, it is not antireflexive.…”

“…Buchberger's algorithm was designed by using techniques which are very similar to the work of Knuth and Bendix, [22]. The details of this approach can be found in Winkler's thesis and in two early articles by Buchberger, [5] and [4]. More recently, a more algebraic presentation of the algorithm appeared.…”

Difference algebra and system identification

“…Algorithmic aspects of Ritt's algorithm are discussed in Diop and Fliess (1991a), Diop and Fliess (1991b), Diop (1989), Diop (1991), Glad (1992) and Glad (1997). For nondifferential problems an alternative to Ritt's theory of characteristic sets is given by Gröbner bases, see, for instance, Buchberger (1976) and Becker and Weispfenning (1993). A comparison of the two approaches can be found in Forsman (1991, Chapter 2).…”