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Abstract We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_f$$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of $$\operatorname{Crit}^{\geqslant 0}_f$$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_{k,n}$$ is homeomorphic to the second hypersimplex $$\Delta _{2,n}$$.
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Real Tropicalization and Analytification of Semialgebraic Sets
Abstract Let $$K$$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $$K$$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $$S \subseteq K^n$$ we define a space $$S_r^{{\operatorname{an}}}$$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $$S$$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $$X$$ is an algebraic variety, we show that $$X_r^{{\operatorname{an}}}$$ can be canonically embedded into the real spectrum $$X_r$$ of $$X$$, and we study its relation with the Berkovich analytification of $$X$$.
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- Award ID(s):
- 1855726
- PAR ID:
- 10361862
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2022
- Issue:
- 2
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- p. 928-958
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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