On Monomial Reduction and Polynomial Expressions with Respect to Binomial Ideals

Abstract

Let I ⊂ K[x ₁,…, x _n ] be an ideal and G be the reduced Gröbner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x ₁,…, x _n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm.

Key Words:

• Computational algebraic geometry
• Components of solutions
• Division algorithm
• Polyhedral method

2010 Mathematics Subject Classification:

• Primary: 14Q15
• Secondary
• 65H10
• 68W30
• 13P05

ACKNOWLEDGMENTS

The author is grateful to the anonymous referees for their valuable comments and helpful suggestions.

Notes

Communicated by S. Bazzoni.

Dedicated to Professor Siavash M. Shahshahani.