yr/tropicalBasics

Project.toml

@@ -18,7 +18,6 @@ Polymake = "d720cf60-89b5-51f5-aff5-213f193123e7"
 Preferences = "21216c6a-2e73-6563-6e65-726566657250"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RandomExtensions = "fb686558-2515-59ef-acaa-46db3789a887"
-RecipesBase = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
 Serialization = "9e88b42a-f829-5b0c-bbe9-9e923198166b"
 Singular = "bcd08a7b-43d2-5ff7-b6d4-c458787f915c"
 TOPCOM_jll = "36f60fef-b880-50dc-9289-4aaecee93cc3"
@@ -40,7 +39,6 @@ Polymake = "0.11.8"
 Preferences = "1"
 Random = "1.6"
 RandomExtensions = "0.4.3"
-RecipesBase = "1.2.1"
 Serialization = "1.6"
 Singular = "0.19.0"
 TOPCOM_jll = "0.17.8"

docs/doc.main

@@ -134,7 +134,7 @@
         "CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md",
      ],
      "CommutativeAlgebra/homological_algebra.md",
-     
+
      "Gröbner/Standard Bases" => [
         "CommutativeAlgebra/GroebnerBases/orderings.md",
         "CommutativeAlgebra/GroebnerBases/groebner_bases.md",
@@ -196,10 +196,6 @@
         "AlgebraicGeometry/ToricVarieties/ToricIdealSheaves.md",
         "AlgebraicGeometry/ToricVarieties/BlowdownMorphisms.md",
      ],
-     "Tropical Geometry" => [
-        "AlgebraicGeometry/TropicalGeometry/intro.md",
-        "AlgebraicGeometry/TropicalGeometry/curve.md",
-     ],
      "Curves" => [
          "AlgebraicGeometry/Curves/AffinePlaneCurves.md",
          "AlgebraicGeometry/Curves/ProjectiveCurves.md",
@@ -241,6 +237,19 @@
    ],
 
 
+  "Tropical Geometry" => [
+     "TropicalGeometry/intro.md",
+     "TropicalGeometry/semiring.md",
+     "TropicalGeometry/semiring_map.md",
+     "TropicalGeometry/variety.md",
+     "TropicalGeometry/hypersurface.md",
+     "TropicalGeometry/curve.md",
+     "TropicalGeometry/linear_space.md",
+     "TropicalGeometry/groebner_theory.md",
+     "TropicalGeometry/tropicalization.md",
+  ],
+
+
   "References" => "references.md",
   "Index" => "manualindex.md",
   "Developer Documentation" => [

docs/oscar_references.bib

@@ -233,6 +233,21 @@ @Article{BJRR12
   doi           = {10.1063/1.3677646}
 }
 
+@Article{BJSST07,
+  author        = {Bogart, T. and Jensen, A. N. and Speyer, D. and Sturmfels, B. and Thomas, R. R.},
+  title         = {Computing tropical varieties},
+  zbl           = {1121.14051},
+  journal       = {J. Symb. Comput.},
+  fjournal      = {Journal of Symbolic Computation},
+  volume        = {42},
+  number        = {1-2},
+  pages         = {54--73},
+  year          = {2007},
+  doi           = {10.1016/j.jsc.2006.02.004},
+  language      = {English},
+  zbmath        = {5203512}
+}
+
 @Article{BKR20,
   author        = {Böhm, Janko and Keicher, Simon and Ren, Yue},
   title         = {Computing {GIT}-fans with symmetry and the {M}ori chamber decomposition of $\overline M_{0,6}$},
@@ -1343,6 +1358,21 @@ @Book{Loo84
   year          = {1984}
 }
 
+@Article{MR20,
+  author        = {Markwig, Thomas and Ren, Yue},
+  title         = {Computing tropical varieties over fields with valuation},
+  zbl           = {1445.14085},
+  journal       = {Found. Comput. Math.},
+  fjournal      = {Foundations of Computational Mathematics},
+  volume        = {20},
+  number        = {4},
+  pages         = {783--800},
+  year          = {2020},
+  doi           = {10.1007/s10208-019-09430-2},
+  language      = {English},
+  zbmath        = {7244216}
+}
+
 @Book{MS05,
   author        = {Miller, Ezra and Sturmfels, Bernd},
   title         = {Combinatorial commutative algebra},

docs/src/AlgebraicGeometry/intro.md

@@ -11,7 +11,6 @@ The algebraic geometry part of OSCAR provides functionality for dealing with
 * schemes
 * toric varieties,
 * toric schemes,
-* tropical geometry.
 
 Computations in affine and projective algebraic geometry rely on [Commutative Algebra](@ref).
 

docs/src/TropicalGeometry/curve.md

@@ -0,0 +1,19 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Tropical curves
+
+## Introduction
+A tropical curve is a graph with multiplicities on its edges.  If embedded, it is a polyhedral complex of dimension (at most) one.
+
+## Construction
+In addition to converting from `TropicalVariety`, objects of type `TropicalCurve` can be constructed from:
+```@docs
+```
+
+## Properties
+In addition to the properties inherited from `TropicalVariety`, objects of type `TropicalCurve` have the following exclusive properties:
+```@docs
+graph(tc::TropicalCurve)
+```

docs/src/TropicalGeometry/groebner_theory.md

@@ -0,0 +1,16 @@
+# Groebner theory
+
+## Introduction
+Groebner bases and related notions in tropical geometry are a generalization of its classical counterparts to fields with valuation.  The Groebner bases take the valuation into account, the initial ideals live over the residue field, and consequently the Groebner complex is not necessarily a polyhedral fan.  For details, see
+- Chapter 2.4 and 2.5 in [MS15](@cite)
+
+## Groebner bases
+```@docs
+groebner_basis(I::MPolyIdeal, nu::TropicalSemiringMap, w::Vector{<:Union{QQFieldElem,ZZRingElem,Rational,Integer}})
+```
+
+## Initial forms and initial ideals
+```@docs
+initial(f::MPolyRingElem, nu::TropicalSemiringMap, w::Vector{<:Union{QQFieldElem,ZZRingElem,Rational,Integer}})
+initial(I::MPolyIdeal, nu::TropicalSemiringMap, w::Vector{<:Union{QQFieldElem,ZZRingElem,Rational,Integer}}; skip_groebner_basis_computation::Bool=false)
+```

docs/src/TropicalGeometry/hypersurface.md

@@ -0,0 +1,25 @@
+# Tropical hypersurfaces
+
+## Introduction
+A tropical hypersurface is a balanced polyhedral complex of codimension one.  It is dual to a regular subdivision of a Newton polytope. For more on tropical hypersurfaces, see
+- Chapter 3.1 in [MS15](@cite)
+- Chapter 1 in [Jos21](@cite)
+
+Objects of type `TropicalHypersurface` need to be embedded, abstract tropical hypersurfaces are currently not supported.
+
+## Constructors
+In addition to converting from `TropicalVariety`, objects of type `TropicalHypersurface` can be constructed from:
+- polynomials over a tropical semiring,
+- polynomials over a field and a tropical semiring map,
+- subdivision of points and a choice of min- or max-convention.
+```@docs
+tropical_hypersurface
+```
+
+## Properties
+In addition to the properties inherited from `TropicalVariety`, objects of type `TropicalHypersurface` have the following exclusive properties:
+```@docs
+algebraic_polynomial(TropH::TropicalHypersurface)
+tropical_polynomial(TropH::TropicalHypersurface)
+dual_subdivision(TropH::TropicalHypersurface)
+```

docs/src/TropicalGeometry/intro.md

@@ -0,0 +1,22 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Introduction
+The tropical geometry part of OSCAR provides functionality for
+- tropical min-plus and max-plus semirings,
+- matrices and polynomials thereover,
+- tropical varieties, hypersurfaces, curves, and linear spaces.
+
+
+General textbooks offering details on theory and algorithms include:
+- Maclagan, Sturmfels: Introduction to Tropical Geometry [MS15](@cite)
+- Joswig: Essentials of Tropical Combinatorics [Jos21](@cite)
+
+Like [MS15](@cite), OSCAR follows the min-convention by default and unless specified otherwise.
+
+## Contact
+Please direct questions about this part of OSCAR to:
+* [Yue Ren](https://www.yueren.de/).
+
+You can also ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack) or raise an issue on [github](https://www.oscar-system.org/community/#how-to-report-issues).

docs/src/TropicalGeometry/linear_space.md

@@ -0,0 +1,31 @@
+# Tropical linear spaces
+
+## Introduction
+A tropical linear space is a balanced polyhedral complex supported on a finite intersection of linear tropical hypersurfaces with all multiplicities one.  It is dual to a matroid subdivision of a hypersimplex, and may arise as tropicalizations of linear ideals. For more on tropical linear spaces, see
+- Chapter 4.4 in [MS15](@cite)
+- Chapter 10 in [Jos21](@cite)
+
+Objects of type `TropicalLinearSpace` need to be embedded, abstract tropical linear spaces are currently not supported.
+
+
+## Constructors
+In addition to converting from `TropicalVariety`, objects of type `TropicalLinearSpace` can be constructed from:
+- Pluecker vectors over a tropical semiring,
+- Pluecker vectors over a field and a tropical semiring map,
+- matrices over a tropical semiring,
+- matrices over a field and a tropical semiring map.
+```@docs
+tropical_linear_space
+```
+
+## Properties
+In addition to the properties inherited from `TropicalVariety`, objects of type `TropicalLinearSpace` have the following exclusive properties:
+```@docs
+pluecker_indices(TropL::TropicalLinearSpace)
+tropical_pluecker_vector(TropL::TropicalLinearSpace)
+algebraic_pluecker_vector(TropL::TropicalLinearSpace)
+tropical_semiring_map(TropL::TropicalLinearSpace)
+tropical_matrix(TropL::TropicalLinearSpace)
+algebraic_matrix(TropL::TropicalLinearSpace)
+algebraic_ideal(TropL::TropicalLinearSpace)
+```

docs/src/TropicalGeometry/semiring.md

@@ -0,0 +1,32 @@
+# Tropical semirings, matrices, and polynomials
+
+## Introduction
+In OSCAR, the tropical semiring is either
+- the min-plus semiring $(\mathbb{Q}\cup\{+\infty\},\oplus,\odot)$ with $a\oplus b=\min(a,b)$ and $a\odot b=a+b$,
+- the max-plus semiring $(\mathbb{Q}\cup\{-\infty\},\oplus,\odot)$ with $a\oplus b=\max(a,b)$ and $a\odot b=a+b$.
+
+Whereas tropical semirings in [MS15](@cite) and [Jos21](@cite) are extensions of the real numbers, tropical semirings in OSCAR are an extension of the rational numbers to avoid precision issues.
+
+## Constructor
+Objects of type `TropicalSemiring`, as well as matrices and polynomials thereover, can be constructed as follows:
+
+!!! warning
+    OSCAR disables unicode by default, so zeroes of tropical semirings are printed as "infty" and "-infty" instead of using their proper unicode characters.  To enabled unicode in the current and future sessions, run `allow_unicode(true)`.
+
+
+```@docs
+tropical_semiring()
+```
+
+## Properties
+Objects of type `TropicalSemiring` have the following properties:
+```@docs
+convention(T::TropicalSemiring{typeof(min)})
+```
+
+## Related functions
+Other functions related to `TropicalSemiring`, matrices, and polynomials thereover include:
+```@docs
+det(A::AbstractAlgebra.Generic.MatSpaceElem{<: Oscar.TropicalSemiringElem})
+tropical_polynomial(f::MPolyRingElem, nu::TropicalSemiringMap)
+```

docs/src/TropicalGeometry/semiring_map.md

@@ -0,0 +1,17 @@
+# Tropical semiring maps
+
+## Introduction
+In OSCAR, a `TropicalSemiringMap` is a map $\nu: K\to\mathbb{T}$ from a field $K$ to a tropical semiring $\mathbb{T}$ satisfing
+1. finiteness: $\nu(a)=\pm\infty$ if and only if $a=0$,
+2. multiplicativity: $\nu(a\cdot b)=\nu(a)+\nu(b)$,
+3. superadditivity: $\nu(a\cdot b)\geq\min(\nu(a),\nu(b))$ (in the order defined in Section 2.7 of [Jos21](@cite)).
+
+Most commonly, $\nu(a)=-\mathrm{val}(a)$ if $\mathbb{T}$ is the min-plus semiring, and $\nu(a)=+\mathrm{val}(a)$ if $\mathbb{T}$ is the max-plus semiring, for some valuation $\mathrm{val}:K^\ast\rightarrow\RR$.  Essentially, $\nu$ captures a valuation on $K$ as well as a choice of min- or max-convention.  They are an optional input for most tropical functions over valued fields (the default being the trivial valuation and the min-convention).
+
+## Constructor
+Tropical semiring maps can be constructed as follows:
+```@docs
+tropical_semiring_map(K::Field, minOrMax::Union{typeof(min),typeof(max)}=min)
+tropical_semiring_map(K::QQField, p::Union{RingElem,Integer,Rational}, minOrMax::Union{typeof(min),typeof(max)}=min)
+tropical_semiring_map(Kt::Generic.RationalFunctionField, t::Generic.RationalFunctionFieldElem, minOrMax::Union{typeof(min),typeof(max)}=min)
+```