First steps in tropical geometry

By Jürgen Richter-Gebert, Bernd Sturmfels, Thorsten Theobald. 2003

441 citations recorded by the source corpus.

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.

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

View Paper: https://www.semanticscholar.org/paper/1b31c0bd5c5eb397953afa1c357b67475aa0637e

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Citations in the corpus

Enumerative tropical algebraic geometry in R^2 (2003, background): “Similarly we can set up the problem of counting all (not necessarily irreducible) curves.”
Non-archimedean amoebas and tropical varieties (2004, background): “Then f t is a convex piecewise-linear function on R known as the tropicalization of f (see [21] and [24] for background).”
Amoebas of Algebraic Varieties and Tropical Geometry (2004, result): “As is was suggested in [ 36 ] non-Archimedean amoebas provide a byway for such definition as tropical varieties can be simply defined as non-Archimedean amoebas for algebraic varieties in (K ∗ ) n .”
A Tropical Toolkit (2006, background): “The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory.”
Supertropical matrix algebra (2008, background): “• The tropical determinant of a matrix A is a ghost iff the rows of A are tropically dependent, iff the columns of A are tropically dependent.”
AFFINE BUILDINGS AND TROPICAL CONVEXITY (2007, background): “the tropical prevariety [ 20 ] specified by the set of all quadratic Pl¨ucker relations.”
The q-theory of Finite Semigroups (2008, background): “The theory of commutative idempotent semirings is a part of what is called tropical geometry, and is currently a hot research topic [293].”
The group law on a tropical elliptic curve (2004, background): “Although perhaps not as enlightening as the homotopy argument given in [6], one can prove Theorem 4.”
Enumerative tropical algebraic geometry (2003, background): “This implication is a version of the so-called Viro patchworking [24] in real algebraic geometry.”
Tropical Geometry and its applications (2006, neutral): “The semifield T is closely related to non-Archimedean fields K with (real) valuations val : K → T.”
First steps in tropical intersection theory (2007, background): “We can summarize this example as follows: Firstly, for a tropical polynomial f , the associated Weil divisor f · R n coincides with the locus of non-differentiability T ( f ) of f (see [ 7, section 3 ]), and secondly, “the k-fold self-intersection of a tropical hyperplane in R n ”i s”