Generic root counts and flatness in tropical geometry

Helminck, Paul Alexander; Ren, Yue

Abstract

We use tropical and nonarchimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space 𝑌. In particular, we are interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes 𝑋 ⊆ 𝑇 where 𝑇 is a relative torus over𝑌. We generalize Bernstein’s theorem from an intersecting family of hypersurfaces 𝑋 = 𝑉(𝑓1 ) ∩ ⋯ ∩ 𝑉(𝑓𝑛 ) to an intersecting family of higher-codimensional schemes𝑋 = 𝑋 1 ∩ ⋯ ∩ 𝑋 𝑘, replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of 𝑋 around a point 𝑃 ∈𝑌, which allows us to transfer tropical properties of the fiber over 𝑃 to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification 𝑌 an, and that the tropical intersection number is attained as a root count at all𝑃 ∈ 𝑌 an around which the 𝑋 𝑖’s are tropically flat and the tropical prevariety of the fibers ⋂𝑘𝑖=1 trop(𝑋 𝑖,𝑃 ) is bounded. We then study the generic root count of a wide class of parametrized square polynomial systems. This, in particular, gives tropical formulas for the volumes of Newton–Okounkov bodies, and the number of complex steady states of chemical reaction networks.

Citation

Helminck, P. A., & Ren, Y. (2025). Generic root counts and flatness in tropical geometry. Journal of the London Mathematical Society, 111(5), e70171. https://doi.org/10.1112/jlms.70171

• Journal Article Type: Article
• Acceptance Date: Apr 9, 2025
• Online Publication Date: May 23, 2025
• Publication Date: 2025-05
• Deposit Date: Jun 6, 2025
• Publicly Available Date: Jun 6, 2025
• Journal: Journal of the London Mathematical Society
• Print ISSN: 0024-6107
• Electronic ISSN: 1469-7750
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 111
• Issue: 5
• Pages: e70171
• DOI: https://doi.org/10.1112/jlms.70171
• Public URL: https://durham-repository.worktribe.com/output/3963539