Buchberger Theory for Effective Associative Rings - Teo Mora
Sprache des Titels:
Englisch
Original Kurzfassung:
The extension of Buchberger Theory and Algorithm from the classical case of polynomial rings over a field[1, 2, 3] to the case of (non necessarily commutative) monoid rings over a (non necessarily free)
monoid and a principal ideal ring was immediately performed by a series of milestone papers: Zacharias? [9] approach to canonical forms, Spear?s[7] theorem which extends Buchberger Theory to each effectively given rings, Möller?s[5] reformulation of
Buchberger Algorithm in terms of lifting. Since the universal property of the free monoid ring Q := Z[Z*] over Z and the monoid Z* of all words over the alphabet Z grants that each ring with identity A can be presented as a quotient A = Q/I of a free
monoid ring Q modulo a bilateral ideal I in Q, in order to impose a Buchberger Theory over any effectiveassociative ring it is sufficient to reformulate it in filtration-valuation terms [8,4, 6] and apply the results quoted above; in particular Zacharias
canonical forms allow to effectively present A and its elements, Spear?s theorem describes how Q imposes its natural filtration on A and a direct application of Möller?s lifting theorem to such filtration allows to characterize the required S-polynomials.
References: [1] Buchberger B., Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph. D. Thesis, Innsbruck (1965) [2] Buchberger B., Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen
Gleischunssystem, Aeq. Math. 4 (1970), 374?38 [3] Buchberger B., Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, in Bose N.K. (Ed.) Multidimensional Systems Theory (1985), 184? 232, Reider [4] T. Mora, Seven variations on standard bases, (1988)
ftp://ftp.disi.unige.it/person/MoraF/PUBLICATIONS/7Varietions.tar.gz
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