Resultants over commutative idempotent semirings I: Algebraic aspect

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• 作者：Hoon Hong^a ; ^{hong@ncsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Yonggu Kim^b ; ^{kimm@jnu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Georgy Scholten^a ; ^{ghscholt@ncsu.edu" class="auth_mail" title="E-mail the corresponding author} ; J. Rafael Sendra^c ; ^{rafael.sendra@uah.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Resultant ; Commutative idempotent semiring ; Tropical algebra ; Sylvester matrix ; Permanent
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：285-308
• 全文大小：415 K

文摘

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.