Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Volumn , Issue , 2009, Pages 151-158

  Faugère, Jean Charles ^a   Rahmany, Sajjad ^a  

^a Centre Paris Rocquencourt   (France)

Author keywords

F5 algorithm; Gr ebner basis; Invariant ring; Sagbi groebner; Symmetric polynomials

Indexed keywords

COMPLEX ROOTS; EFFICIENT METHOD; ELEMENTARY SYMMETRIC POLYNOMIAL; FINITE GROUPS; GENERALISATION; INVARIANT POLYNOMIALS; INVARIANT RING; POLYNOMIAL EQUATION; POLYNOMIAL RINGS; POLYNOMIAL SYSTEMS; SUBALGEBRAS; SYMMETRIC POLYNOMIAL; ZERO-DIMENSIONAL IDEAL;

ALGORITHMS; COMPUTATION THEORY; POLYNOMIALS;

COMPUTATIONAL EFFICIENCY;

EID: 77950433180     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1576702.1576725     Document Type: Conference Paper

References (18)

1. A.Colin. Solving a system of algebraic equations with symmetries. Pure and applied algebra, 117-118:195-215, 1997.
2. G. A.Conca, J.Herzog. Sagbi bases and application to blow-up algebras. Reine Angew.Math, pages 113-138, 1996.
3. n, whose Fourier transforms have constant modulus, and "cyclic n-roots". In J.S. Byrnes and J.F. Byrnes, editor, Recent Advances in Fourier Analysis and its Applications, volume 315 of Ser. C: Math. Phys. Sci., Kluwer, pages 131-140. NATO Adv. Sci. Inst., 1989.
4. B.Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, Wien,New York, 1993.
5. Buchberger B. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck, 1965.
6. Buchberger B. An Algorithmical Criterion for the Solvability of Algebraic Systems. Aequationes Mathematicae, 4(3):374-383, 1970. (German).
7. Buchberger B. A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner Basis. In Proc. EUROSAM 79, volume 72 of Lect. Notes in Comp. Sci., pages 3-21. Springer Verlag, 1979.
8. D. Cox, J. Little, and D. O'Shea. Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer Verlag, New York, 3rd ed. 2007. corr. 2nd printing, 2008, xvi edition, 2007. 560 p.
9. Faugère J.C. A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra, 139(1-3):61-88, June 1999.
10. Faugère J.C. A new efficient algorithm for computing Gröbner bases without reduction to zero F5. In T. Mora, editor, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pages 75-83. ACM Press, July 2002.
11. Faugère, J.C., Gianni, P., Lazard, D. and Mora T. Efficient Computation of Zero-Dimensional Gröbner Basis by Change of Ordering. Journal of Symbolic Computation, 16(4):329-344, October 1993.
12. G.Kapur and K.Madlener. A completion procedure for computing a canonical basis for a k-subalgebra. Computers and mathematics, pages 1-11, June 1989.
13. G.Kemper. Computational Invariant Theory. Springer-Verlag, New York, 2002.
14. J. Hoffstein, J. Pipher, and J. Silverman. Ntru: a ring-based public key cryptosystem. In ANTS III, volume 1423, pages 267-288. Springer Verlag, 1998.
15. J.L.Miller. Analogues of Gröbner bases in polynomial rings over a ring. Journal Of Symbolic Computation, 21(2):139-153, June 1996.
16. J.L.Miller. Effective algorithms for intrinsically computing SAGBI-Gröbner bases in a polynomial ring over a filed. Groebner bases and application (Linz), pages 421-433, February 1998.
17. L.Robbiano and M. Sweedler. Subalgebra bases. Commutative algebra, pages 61-87, 1990.
18. N.M. Thiéry. Computing minimal generating sets of invariant rings of permutation groups with SAGBI-Gröbner basis. International Conference DM-CCG, Discrete Model-Combinatorics, Computation and Geometry, pages 84-89, July 2002.