Congruences on tropical rational function semifields and tropical curves

2023/05/07 by JuAe Song, Song, JuAe · 1 citation
Computer Science · Mathematics · #Polynomial and algebraic computation #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2305.04204

Abstract

We define tropical rational function semifields \boldsymbolT(X₁, …, X_n) and prove that a tropical curve \varGamma is realized (except for points at infinity) as the congruence variety V ⊂ \boldsymbolRⁿ associated with a congruence on \boldsymbolT(X₁, …, X_n) by giving a specific map \varGamma → V. Also, we shed light on the relation between congruences E on \boldsymbolT(X₁, …, X_n) and congruence varieties associated with them and reveal the quotient semifield \boldsymbolT(X₁, …, X_n) / E to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry.

Cited by

• Non-finite generatedness of the congruences defined by tropical varieties

Related

• Finitely generated congruences on tropical rational function semifields
• Congruences and coordinate semirings of tropical varieties
• Rational function semifields of tropical curves are finitely generated over the tropical semifield
• Semiring isomorphisms between rational function semifields of tropical curves induce isomorphisms between tropical curves
• A geometric interpretation of Krull dimensions of \boldsymbolT-algebras