We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field $k(t_1,\dots,t_d)$ of rational functions over an algebraically closed field $k$ of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the $d$-connectivity result for tropical varieties from that paper.
Toric and tropical Bertini theorems in prime characteristic*
Ashley K. Wheeler, Georgia TechAuthors: Francesca Gandini, Milena Hering, Diane Maclagn, Fatemeh Mohammadi, Jenna Rajchgot, Ashley K. Wheeler, Josephine Yu
2022 AWM Research Symposium
Homological and Combinatorial Aspects of Commutative Algebra