Computing SAGB-Gröbner Basis of Ideals of Invariant Rings by Using Gaussian Elimination
The link between Gröbner basis and linear algebra was
described by Lazard [4,5] where he realized the Gr¨obner basis
computation could be archived by applying Gaussian elimination over
Macaulay-s matrix .
In this paper, we indicate how same technique may be used to
SAGBI- Gröbner basis computations in invariant rings.
[1] Buchberger B., Ein Algorithmus zum Auffiden der Basiselemente des
Restklassenringes nach einem nulldimensionalen Polynomideal, Innsbruck,
1965
[2] Buchberger B., Ein algorithmisches Kriterium f¨ur die L¨osbarkeit eines
algebraischen Gleichungssystems, Aequationes Math., 4, pages 374-383,
1970.
[3] David A. Cox and John B. Little and Don O-Shea, Ideals, Varieties,
and Algorithms : An introduction to computational algebraic geometry
and commutative algebra,Undergraduate Texts in Mathematics. Springer
Verlag, New York, 3rd ed.2007.
[4] D.Lazard,Gr¨obner bases, Gaussian elimination and resolution of systems
of algebraic equations, Computer algebra (London, 1983), Lecture Notes
in Comput. Sci.162.
[5] D.Lazard,Solving systems of algebric equations, ACM SIGSAM Bulletin,
35, pages 11-37, 2001.
[6] Faug`ere, J-C. and Rahmany, S., Solving systems of polynomial equations
with symmetries using SAGBI-Gr¨obner bases, ISSAC 2009.
[7] F.S. Macaulay On Some formulae in elimination,proceedings of the
London Mathematical Society, 33, page3-27, 1902.
[8] F.S. Macaulay, The Algebaic Theory of Modular Systems, Cambridge
Mathematical Librar, Cambridge University Press, 1916.
[9] J.L.Miller, Analogues of Gr¨obner bases in polynomial rings over a ring,
Journal of Symbolic Computation, 21(2), 139-153, 1996.
[10] J.L.Miller, Effective algorithm for intrinsically computing SAGBIGröbner bases in polynomial ring over a field, Groebner bases and
application (Linz),421-433, 1998.
[1] Buchberger B., Ein Algorithmus zum Auffiden der Basiselemente des
Restklassenringes nach einem nulldimensionalen Polynomideal, Innsbruck,
1965
[2] Buchberger B., Ein algorithmisches Kriterium f¨ur die L¨osbarkeit eines
algebraischen Gleichungssystems, Aequationes Math., 4, pages 374-383,
1970.
[3] David A. Cox and John B. Little and Don O-Shea, Ideals, Varieties,
and Algorithms : An introduction to computational algebraic geometry
and commutative algebra,Undergraduate Texts in Mathematics. Springer
Verlag, New York, 3rd ed.2007.
[4] D.Lazard,Gr¨obner bases, Gaussian elimination and resolution of systems
of algebraic equations, Computer algebra (London, 1983), Lecture Notes
in Comput. Sci.162.
[5] D.Lazard,Solving systems of algebric equations, ACM SIGSAM Bulletin,
35, pages 11-37, 2001.
[6] Faug`ere, J-C. and Rahmany, S., Solving systems of polynomial equations
with symmetries using SAGBI-Gr¨obner bases, ISSAC 2009.
[7] F.S. Macaulay On Some formulae in elimination,proceedings of the
London Mathematical Society, 33, page3-27, 1902.
[8] F.S. Macaulay, The Algebaic Theory of Modular Systems, Cambridge
Mathematical Librar, Cambridge University Press, 1916.
[9] J.L.Miller, Analogues of Gr¨obner bases in polynomial rings over a ring,
Journal of Symbolic Computation, 21(2), 139-153, 1996.
[10] J.L.Miller, Effective algorithm for intrinsically computing SAGBIGröbner bases in polynomial ring over a field, Groebner bases and
application (Linz),421-433, 1998.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64045", author = "Sajjad Rahmany and Abdolali Basiri", title = "Computing SAGB-Gröbner Basis of Ideals of Invariant Rings by Using Gaussian Elimination", abstract = "The link between Gröbner basis and linear algebra was
described by Lazard [4,5] where he realized the Gr¨obner basis
computation could be archived by applying Gaussian elimination over
Macaulay-s matrix .
In this paper, we indicate how same technique may be used to
SAGBI- Gröbner basis computations in invariant rings.", keywords = "Gröbner basis, SAGBI- Gröbner basis, reduction,Invariant ring, permutation groups.", volume = "4", number = "2", pages = "322-4", }
