Symmetric polynomials in upper-bound semirings. (March 2021)

• Record Type: Journal Article
• Authors: Kališnik, Sara; Lešnik, Davorin
• Is Part Of: Journal of symbolic computation. Volume 103(2021)
• Page Start: 280
• Page End: 299
• Publication Date: 2021-03
• Subjects: Tropical semiring -- Extended semiring -- Supertropical semiring -- Elementary symmetric polynomials -- The fundamental theorem of symmetric polynomials; 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.2020.02.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical semirings. These all fall into a larger class of upper-bound semirings. In this paper we extend the known results and give a complete characterization of elementary upper-bound semirings. We further improve this characterization statement in the case of linearly ordered upper-bound semirings.