Resultants over Commutative Idempotent Semirings I: (Algebraic aspect)
题目：Resultants over Commutative Idempotent Semirings I: (Algebraic aspect)
报告人：Hoon Hong（North Carolina State University）
摘要：The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theory has two aspects: algebraic and geometric. In this talk, 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. Recently, we proved that the same property (with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring.
In this talk, we will briefly go over the followings:
1. The well-known resultant theory over commutative rings (Sylvester 1853).
2. An adaptation to the tropical semiring (Odagiri 2008).
3. An extension to supertropical semirings (Izhakian and Rowen 2010).
4. An extension to commutative idempotent semirings (our recent result 2015).
This is a joint work with Yonggu Kim, Georgy Scholten, J. Rafael Sendra.