Resultants over commutative idempotent semirings I: Algebraic aspect. (March 2017)

• Record Type: Journal Article
• Authors: Hong, Hoon; Kim, Yonggu; Scholten, Georgy; Sendra, J. Rafael
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 2
• Page Start: 285
• Page End: 308
• Publication Date: 2017-03
• Subjects: Resultant -- Commutative idempotent semiring -- Tropical algebra -- Sylvester matrix -- Permanent; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.02.009
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theory has two aspects: algebraic and geometric. In this paper, we focus on the algebraic aspect. One of the most important and well known algebraic properties of the resultant is that it is equal to the determinant of the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiring if one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraic structures called commutative idempotent semiring. In this paper, we prove that the same property (with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring.