On solving systems of algebraic equations via ideal bases and elimination theory

ABSTRACT

The determination of solutions of a system of algebraic equations is still a problem for which an efficient solution does not exist. In the last few years several authors have suggested new or refined methods, but none of them seems to be satisfactory. In this paper we are mainly concerned with exploring the use of Buchberger's algorithm for finding Groebner ideal bases [2] and combine/compare it with the more familiar methods of polynomial remainder sequences (pseudo-division) and of variable elimination (resultants) [4].

REFERENCES

• 1: Buchberger, B.: "Ein algorithmisches Kriterium fur die Loesbarkeit eines algebraischen Gleichungs-systems", Aequationes Math. 4, (1969), 374-383.
• 2: B. Buchberger, Some properties of Gröbner-bases for polynomial ideals, ACM SIGSAM Bulletin, v.10 n.4, p.19-24, November 1976 [doi>10.1145/1088222.1088224]
• 3: Trinks, W.: "Ueber B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu loesen", Journal of Number Theory, Vol. 10, No. 4, Nov. 1978, 475-488.
• 4: David Y. Y. Yun, On algorithms for solving systems of polynomial equations, ACM SIGSAM Bulletin [doi>10.1145/1086809.1086811]
• 5: Spear, D. A.: "A Constructive approach to commutative ring theory", Proceedings 1977 MAC-SYMA Users' Conference, NASA CP-2012, July 1977, pp. 369-376.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.3 Languages and Systems

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms

General Terms:
Algorithms, Theory