Enumerative tropical algebraic geometry

By G. Mikhalkin. 2003

398 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1e92522922fab865be68a30f46fd8131ebd7d267

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In Algebra and Number Theory

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Citations in the corpus

The Tropical Rank of a Tropical Matrix (2006, background): “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, 21, 22, 23, 24]; a short brief can be found in [14]. This mathematics takes place”
Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry (2002, background): “…enumerative geometry E.Shustin y A bstract W edem onstratea tropicalapproach to enum eration ofsingularcurveson toric surfaces,which consistsofreducing theenum eration ofalgebraic curves to enum eration ofnon-Archim edean am oebas,the im agesofalgebraic curves byareal-valued non-Archim edean…”
Hyperfields for Tropical Geometry I. Hyperfields and dequantization (2010, background): “ literally the same as the proof of Theorem 7.C above. A ring can be replaced here by an algebraically closed ﬁeld realpower Puiseux series ∑ r∈I a tr, where I ⊂ Ris a well-ordered set. Cf. Mikhalkin [13], Section 6. This construction demonstrates how one can obtain the tropical addition of complex ”
Skeletons and tropicalizations (2014, methodology): “lgebraic proof and a generalization of Mikhalkin’s correspondence theorem. The latter is the key in Mikhalkin’s pioneering work on Gromov–Witten invariants on the plane. Mikhalkin’s original proof in [Mik05] is based on complex analytic and symplectic techniques which are fundamentally different fro”
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”
Tropical Cramer Determinants Revisited (2013, background): “sions have been used more recently. In particular, the incorporation of \phase" (instead of sign) information in tropical constructions has played an important role in the arguments of Mikhalkin [Mik05]. Moreover, Viro introduced a general notion of hypereld [Vir10] with a multivalued addition,”
On the frontiers of polynomial computations in tropical geometry (2004, background): “ of the original motivations for tropical varieties was a combinatorial approach to certain problems from enumerative geometry suggested by Kontsevich, and that program has been realized by Mikhalkin [14]. Tropical varieties are also related to the observation that algebraic varieties have very simp”
Mirror symmetry for toric Fano manifolds via SYZ transformations (2008, background): “ee that, as in the holomorphic case, we have TGW X¯,δ 0,na+2(PD(pt);D1,a,...,Dna+1,a,ti)= ˆ 1 if ti =PD(pt) and δ =δa 0 otherwise. Indeed, as a special case of the correspondence theorem of Mikhalkin [32] and Nishinou-Siebert [38], we have: For a given point p ∈X, let ξ := Log(p) ∈ NR. Then the uniq”
On basic concepts of tropical geometry (2011, methodology): “In particular, to solve problems of enumerative geometry by tropical techniques, Mikhalkin [ 19 ] considered plane complex tropical curves as images of curves over the field of Puiseux series and as the limits of holomorphic curves under appropriate degeneration of the complex structure (see [19, Se”
From real affine geometry to complex geometry (2007, background): “symmetry, at a much deeper level than the usual description in terms of periods. AFFINE AND COMPLEX GEOMETRY 7 Furthermore, our description of the B-model side is tropical in nature. It is well-known [Mk][NiSi] that one should expect a correspondence between tropical curves on B and families of holo”
Limit shapes and the complex Burgers equation (2005, neutral): “ted in one of two ways) leads to the change of the topology of the real locus of Q0 for all large c. There is a c-dependent deformation of Q T for which every quadruple point becomes a real node, see [16]. Thus, for all c ≫ 0, we have constructed a genus zero curve passing through the points (23). T”
Skeletons and Fans of Logarithmic Structures (2015, neutral): “to this volume [Wer15]. 9.2. Improved fans and moduli spaces. One of the primary applications of tropical geometry, and therefore of fans, is through moduli spaces. Mikhalkin’s correspondence theorem [Mik05], Nishinou and Siebert’s vast extension [NS06], the work [CMR] of Cavalieri, Markwig and Rang”
Tropical geometry and correspondence theorems via toric stacks (2010, methodology): “ucture coming from the degeneration of the toric variety. Finally, they proved that under certain conditions the algebraic curve can be reconstructed uniquely from its degeneration as log-variety. In [9, 12, 13], the parameterized tropical curve Γcorresponding to an algebraic curve C was constructed”