Dequantization of Real Algebraic Geometry on Logarithmic Paper

By O. Y. Viro. 2000

198 citations recorded by the source corpus.

On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.

Subfields: Algebra and Number Theory, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6d62d52ab994104a72401f308627fe76e8ad2414

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Citations in the corpus

Logarithmic limit sets of real semi-algebraic sets (2007, methodology): “ed family. We will use the notation V∗ = (Vt)t>0 for the family, and we will call these families patchworking families determined by V, as they are a generalization of the patchworking families of [Vi]. Given a patchworking family V∗, we can deﬁne the tropical limit of the family as: A0(V∗) = lim”
From the sixteenth Hilbert problem to tropical geometry (2008, background): “Combinatorial patchworking can be presented as a construction of real tropical plane curve and its dequantization. See details in [ 41 ].”
Inflection Points of Real and Tropical Plane Curves (2011, methodology): “ng also provides a systematic method to construct maximally inﬂected real curves. For the sake of shortness we do not recall this technique here, we refer instead to the tropical presentation made in [Vir01], [Mik04], or [Bru09]. In non-tropical terms, Theorem 1 states that any real primitive T-curv”
Linear independence over tropical semirings and beyond (2008, background): “Max-plus structures have appeared in relation with in a variety of fields, like operations research and optimization [Vor67, CG79, GM08b], discrete event systems [BCOQ92, CGQ99], automata theory (see [Pin98] and its references), quasi-classic asymptotics [Mas87], Hamilton-Jacobi partial differential”
Logarithmic equivalence of Welschinger and Gromov-Witten invariants (2004, unknown): “(1)”
The Tropical Rank of a Tropical Matrix (2006, result): “n-singular minor of size k×k. Concerning matrices over R, this setting agrees with that of Develin, Santos and Sturmfels [3]. The notion of tropical mathematics was introduced only in the past decade [5, 26], since then this theory has developed rapidly and led to many applications [4, 6, 13, 15, 16”
Tropical Cramer Determinants Revisited (2013, methodology): “Ideas of this nature were indeed essential in the development by Viro of the patchworking method (see the references in [Vir01]), as well as in the construction by Plus [Plu90] of the symmetrized tropical (or max-plus) semiring, Smax.”
On the frontiers of polynomial computations in tropical geometry (2004, background): “Tropical varieties are also related to the observation that algebraic varieties have very simple behavior at infinity when plotted on “log paper” [2, 28 ].”
Formes différentielles réelles et courants sur les espaces de Berkovich (2012, unknown): “Introduit initialement dans le cas où k est le corps des nombres complexes et X est une variété algébrique (voir [6, 56, 47]), ce point de vue a ensuite été développé dans le cas ultramétrique (citons notamment [10, 33, 39]).”
Idempotent Functional Analysis: An Algebraic Approach (2000, background): “Recently O. Viro [ 76 ] described the result of idempotent dequantization of real algebraic geometry.”
First steps in tropical geometry (2003, background): “For the relationship to Maslov dequantization see [ 20 ].”
Toric reduction and a conjecture of Batyrev and Materov (2003, methodology): “Z(ξ) in U for any generic ξ. The proof uses certain type of degenerations reminiscent of the methods of tropical geometry in real algebraic geometry (cf. [ 19 , 18]).”
Tropical Geometry and its applications (2006, unknown): “Indeed, we have logt (ab) = “ logt (a) logt (b)” and “ logt (a) + logt (b)” ≤ logt (a + b) ≤ “ logt (a) + logt (b)” + logt 2 by elementary considerations.”
Brief introduction to tropical geometry (2015, background): “eometry for any cycle of C, if e 1;:::;e kare the edges in ∩T, and if (u i;v i)is a primitive integer vector in the direction of e i, then k Q i=1 (u i;v i)=0 mod 2: (1) The Viro patchworking theorem [Vir01] may be reformulated in terms of twist-admissible sets as follows. Theorem 3.4. For any twist”
Moduli spaces of rational tropical curves (2007, background): “These operations may be viewed as a result of the so-called dequantization of the classical arithmetic operations that underlies the patchworking construction, see [3] and [8].”
Amoebas of Algebraic Varieties and Tropical Geometry (2004, unknown): “Let Log t : (C ∗ ) n → R n be defined by (x1, . . . , xn) = (log |z1|, . . . ,log |zn|). Proposition 4.3 (Maslov [25],Viro [ 43 ]).”
On basic concepts of tropical geometry (2011, background): “There is a continuous deformation that transforms the semifield R≥0 of nonnegative real numbers with ordinary operations of addition and multiplication into the tropical semifield T (see [15, 29 ]).”