Abstract
We present some interesting computational applications of Macaulay’s notion of inverse systems and Noether equations. In particular, we discuss an algorithm by Macualay which computes the forgotten notion (introduced by Emmy Noether) of reduced irreducible decomposition for ideals of the polynomial ring.
Similar content being viewed by others
References
Alonso, M.E., Marinari, M.G., Mora, T.: The big Mother of all dualities: Möller Algorithm. Comm. Alg. 31(2), 783–818; 374–383 (2003)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph. D. Thesis, Innsbruck (1965)
Buchberger B. (1970) Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleischunssystem. Aeq. Math. 4, 374–383
Buchberger B., Möller H.M. (1982) The construction of Multivariate polynomials with Preassigned zeros. Lecture Notes Comput. Sci. 144, 24–31
Cerlienco, L., Mureddu, M.: Algoritmi combinatori per l’interpolazione polinomiale in dimensione ≥ 2. Preprint (1990)
Cerlienco L., Mureddu M. (1995) From algebraic sets to monomial linear bases by means of combinatorial algorithms. Discrete Math. 139, 73–87
Gao, S., Rodrigues, V.M., Stroomer, J.: Gröbner basis structure of finite sets of points. (preprint)
Gianni P. (1987) Properties of Gröbner bases under specialization. Lecture Notes Comput. Sci. 378, 293–297
Gröbner W. (1949) Moderne Algebraische Geometrie. Bibliographisches Institut Mannheim. Springer, Berlin Heidelberg, New York
Kalkbrener M. (1987) Solving systems of Algebraic equations by using Gröbner bases. Lecture Notes Comput Sci. 378, 282–292
Lazard D. (1985) Ideal basis and primary decomposition: case of two variables. J. Symb. Comput. 1, 261–270
Macaulay F.S. (1913) On the resolution of a given modular system into primary systems including some properties of Hilbert numbers. Math. Ann. 74, 66–121
Macaulay F.S. (1916) The Algebraic theory of modular systems. Cambridge University Press, Cambridge
Marinari M.G., Möller H.M., Mora T. (1993) Gröbner bases of ideals defined by functionals with an application to ideals of projective points. AAECC 4, 1–45
Marinari M.G., Möller H.M., Mora T. (1996) On multiplicities in polynomial system solving. Trans. AMS, 348:3283–3321
Marinari M.G., Mora T. (2003) A remark on a remark by Macaulay or Enhancing Lazard Structural Theorem. Bull. Iranian Math. Soc. 29(1):103–145
Marinari, M.G., Mora, T.: Some comments on Cerlienco–Mureddu algorithm and Enhanced Lazard Structural Theorem rejected by ISSAC-2004 (2004)
Marinari, M.G., Mora, T.: Cerlienco–Mureddu Correpondence and Lazard Structural Theorem. Investigaciones Mathematicas (Submitted)(2004)
Möller H.M. (1993) Systems of Algebraic equations solved by means of endomorphisms. Lecture Notes Comput. Sci. 673, 43–56
Noether E. (1921) Idealtheorie in Ringbereichen. Math. Annalen 83, 25–66
Author information
Authors and Affiliations
Corresponding author
Additional information
M. E. Alonso: partially supported by EU Project HPRN-CT-2002-00271 and Spanish Project GEOR-2005-02685.
M. G. Marinari partially supported by MURST and GNSAGA.
T. Mora partially supported by MURST and GNSAGA.
Rights and permissions
About this article
Cite this article
Alonso, M.E., Marinari, M.G. & Mora, T. The big Mother of all dualities 2: Macaulay bases. AAECC 17, 409–451 (2006). https://doi.org/10.1007/s00200-006-0019-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-006-0019-4