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New constructive methods in classical ideal theory. (English) Zbl 0621.13007
Gröbner bases are generalized for submodules of a finite free module over a polynomial algebra P over a field and some algorithms to compute them are also given. These show a possible way to construct effectively a minimal resolution for a P-module. It is also presented a different way which is applied to the computation of the Hilbert function. This technique allows to determine the length, ranks and degrees of a resolution without really constructing it.

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
13-04 Software, source code, etc. for problems pertaining to commutative algebra
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[1] Bayer, D.A, The division algorithm and the Hilbert scheme, ()
[2] Buchberger, B, Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal, () · Zbl 1245.13020
[3] Buchberger, B, Ein algorithmisches kriterium für die Lösbarkeit eines algebraischen gleichungssystems, Aequaliones math., 4, 364-383, (1970) · Zbl 0212.06401
[4] Buchberger, B, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM bull., Vol. 39, 19-29, (1976)
[5] Buchberger, B, A criterion for deterting unnecessary reductions in the construction of Gröbner bases, (), 3-21
[6] Buchberger, B, Gröbner bases: an algorithmic method in polynomial ideal theory, () · Zbl 0587.13009
[7] Buchberger, B, A note on the complexity of constructing Gröbner bases, (), 137-145 · Zbl 0539.13001
[8] Bachmair, L; Buchberger, B, A simplified proof of the characterization theorem for Gröbner bases, (), 29-34 · Zbl 0454.68021
[9] Buchberger, B; Winkler, F, Miscellaneous results on the construction of Gröbner bases for polynomial ideals I, () · Zbl 0607.03003
[10] Galligo, A, A propos du théorème de préparation de wieierstrass, (), 543-579
[11] Galligo, A, Théorème de division et stabilité en géométrie analytique locale, Ann. inst. Fourier, 29, 107-184, (1979) · Zbl 0412.32011
[12] Giusti, M, Some effectivity problems in polynomial ideal theory, (), 159-171 · Zbl 0585.13010
[13] Guiver, J.P, Contribution to two dimensional systems theory, ()
[14] Hironaka, H, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of math., 79, 109-326, (1964) · Zbl 0122.38603
[15] Hermann, G, Die frage der endlich vielen schritte in der theorie der polynomideale, Math. ann., 95, 736-788, (1926) · JFM 52.0127.01
[16] Kollreider, C, Polynomial reduction: the influence of the ordering of terms on a reduction algorithm, () · Zbl 0417.68028
[17] Lazard, D, Commutative algebra and computer algebra, (), 40-48 · Zbl 0552.68047
[18] Lazard, D, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, (), 146-156
[19] Macaulay, F.S, Algebraic theory of modular systems, Cambridge tracts in math. and math. phys., Vol. 19, (1916) · Zbl 0802.13001
[20] Macaulay, F.S, Some properties of enumeration in the theory of modular systems, (), 531-555 · JFM 53.0104.01
[21] Möller, H.M; Mora, F, Upper and lower bounds for the degree of Gröbner bases, (), 172-183
[22] Mora, F, An algorithm to compute the equations of tangent cones, (), 158-165
[23] Northcott, D.G, Lessons on rings, modules and multiplicities, (1968), Cambridge Univ. Press London · Zbl 0159.33001
[24] Renschuch, B, Elementare und praktische idealtheorie, () · Zbl 0354.13001
[25] Schreyer, F.-O, Die berechnung von syzygien mit dem verallgemeinerten Weierstrass’schen divisionsatz…, ()
[26] Sperner, E, Über einen kombinatorischen satz von Macaulay, (), 149-163 · JFM 55.0663.01
[27] Spear, D.A, A constructive approach to commutative ring theory, (), 369-376
[28] Trinks, W, Über B. buchbergers verfahren systeme algebraischer gleichungen zu lösen, J. number theory, 10, 475-488, (1978) · Zbl 0404.13004
[29] Winkler, F, On the complexity of the Gröbner bases algorithm over K[X, Y, Z], (), 184-194
[30] Zacharias, G, Generalized Gröbner bases in commutative polynomial rings, ()
[31] Bayer, D.A, The division algorithm and the Hilbert scheme, ()
[32] Buchberger, B, A criterion for detecting unnecessary reductions in the constructions of Gröbner bases, (), 3-21 · Zbl 0417.68029
[33] de Concini, C; Eisenbud, D; Procesi, C, Hodge algebras, Asterisque, 91, (1982) · Zbl 0509.13026
[34] Gröbner, W, Moderne algebraische geometrie, (1949), Springer-Verlag Vienna · Zbl 0033.12706
[35] Ostrowski, A, Über ein algebraisches übertragungsprinzip, (), 281-326 · JFM 48.0106.03
[36] Renschuch, B, Elementare und praktische idealtheorie, (1976), VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0354.13001
[37] Schreyer, F.-O, Die berechnung von syzygien mit dem verallgemeinerten Weierstrass’schen divisionsatz…, ()
[38] Spear, D.A, A constructive approach to commutative ring theory, (), 369-376
[39] Stanley, R.P, Hilbert functions of graded algebras, Advanc. in math., 28, 57-83, (1978) · Zbl 0384.13012
[40] Zacharias, G, Generalized Gröbner bases in commutative polynomial rings, ()
[41] Buchberger, B, Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal, () · Zbl 1245.13020
[42] Gröbner, W, ()
[43] Hilbert, D, Über die theorie der algebraischen formen, Math. ann., 36, 473-534, (1890) · JFM 22.0133.01
[44] Lazard, D, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, () · Zbl 0539.13002
[45] Mora, F; Möller, H.M, The computation of the Hilbert function, () · Zbl 0575.13006
[46] \scH. M. Möller and F. Mora, Computational aspects of reduction strategies to construct resolutions of monomial ideals, in “Proceedings of AAECC-2” Lect. Notes in Comp. Sci., to appear.
[47] Renschuch, B, Elementare und praktische idealtheorie, (1976), VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0354.13001
[48] Trinks, W, Über B. buchbergers verfahren systeme algebraischer gleichungen zu lösen, J. number theory, 10, 475-488, (1978) · Zbl 0404.13004
[49] Zariski, O; Samuel, P, (), Reprinted by Springer-Verlag, Berlin, 1975.
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