Resultants over Commutative Idempotent Semirings

TL;DR

This paper generalizes the property of the resultant being equal to the determinant of the Sylvester matrix from tropical semirings to all commutative idempotent semirings, expanding its algebraic applicability.

Contribution

It proves that the property holds over any commutative idempotent semiring, not just tropical semirings, with subtraction replaced by addition.

Findings

The resultant equals the determinant of the Sylvester matrix over these semirings.

The property previously known for tropical semirings is extended to all commutative idempotent semirings.

The proof involves replacing subtraction with addition in the algebraic structures.

Abstract

The resultant plays a crucial role in (computational) algebra and algebraic geometry. One of the most important and well known 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 \emph{arbitrary\/} commutative idempotent semiring.