The purpose of this paper is to introduce some tools and results suitable for the study of high-order evaluation of function fields of algebraic varieties. It is based on a study of higher rank quasi-unary evaluations that take values in lexicographically ordered groups R^k.
As the tangent cone of the dual cone complex, we prove a duality theorem that gives a geometric realization of the higher-rank quasi-unary evaluation. Using this duality, we provide an analytic description of the quasi-unary evaluation as a multi-way differential operator of tropical functions.
Furthermore, by considering a sophisticated notion of tropicalization that remembers the initial terms of the power series in each cone of the dual complex, and denoting any compatible collection of initial terms of the cones, number theory We prove a tropical analogue of the weak approximation theorem in . A hallmark of dual cone complexes is the sophisticated tropicalization of rational functions in the functional field of diversity.
By giving the value group R^k its Euclidean topology, we study natural topologies in the space of high-rank evaluation called tropical topologies. By using an approximation theorem, we provide an explicit description of the tropical topology on the tangent cone of the dual cone complex.
Finally, we show that the tangential cone of the dual complex provides the notion of a skeleton in higher-order non-Archimedean geometries. That is, we generalize the rank-1 figure to higher ranks to create retraction maps to the tangent cones of the dual cone complex and use them to obtain the limit equations. In this formula, we reconstruct higher-rank non-Archimedean spaces with the tropical topology as the projective limit. of their higher-ranking skeletons.
We speculate that these high-rank skeletons provide a suitable basis for the study of Newton–Okunkoff body variations.