experimental: Gröbner Walk (#3821)

Co-authored-by: Ortfs <[email protected]>
Co-authored-by: Lars Göttgens <[email protected]>

Author: 3 people

docs/oscar_references.bib

@@ -37,6 +37,30 @@ @Article{AG10
   eprint        = {0810.1148}
 }
 
+@InProceedings{AGK96,
+  author        = {Amrhein, Beatrice and Gloor, Oliver and Küchlin, Wolfgang},
+  title         = {Walking faster},
+  booktitle     = {Design and Implementation of Symbolic Computation Systems},
+  series        = {DISCO 1996},
+  address       = {Heidelberg, Germany},
+  publisher     = {Springer Berlin, Heidelberg},
+  pages         = {150--161},
+  year          = {1996},
+  doi           = {10.1007/3-540-61697-7_14},
+  location      = {Karlsruhe, Germany}
+}
+
+@Article{AGK97,
+  author        = {Amrhein, Beatrice and Gloor, Oliver and Küchlin, Wolfgang},
+  title         = {On the walk},
+  journal       = {Theoretical Computer Science},
+  volume        = {187},
+  number        = {1},
+  pages         = {179--202},
+  year          = {1997},
+  doi           = {10.1016/s0304-3975(97)00064-9}
+}
+
 @Article{AHK18,
   author        = {Adiprasito, Karim and Huh, June and Katz, Eric},
   title         = {Hodge theory for combinatorial geometries},
@@ -453,6 +477,27 @@ @Article{CHM98
   doi           = {10.1112/S1461157000000115}
 }
 
+@Article{CKM97,
+  author        = {Collart, S. and Kalkbrener, M. and Mall, D.},
+  title         = {Converting Bases with the Gröbner Walk},
+  journal       = {Journal of Symbolic Computation},
+  volume        = {24},
+  number        = {3},
+  pages         = {465--469},
+  year          = {1997},
+  doi           = {10.1006/jsco.1996.0145}
+}
+
+@Book{CLO05,
+  author        = {Cox, David A. and Little, John and O'Shea, Donal},
+  title         = {Using Algebraic Geometry},
+  series        = {Graduate Texts in Mathematics},
+  volume        = {185},
+  publisher     = {Springer-Verlag},
+  year          = {2005},
+  doi           = {10.1007/b138611}
+}
+
 @Book{CLS11,
   author        = {Cox, David A. and Little, John B. and Schenck, Henry K.},
   title         = {Toric varieties},
@@ -910,6 +955,17 @@ @Article{FGLM93
   doi           = {10.1006/jsco.1993.1051}
 }
 
+@Article{FJLT07,
+  author        = {Fukuda, K. and Jensen, A. N. and Lauritzen, N. and Thomas, R.},
+  title         = {The generic Gröbner walk},
+  journal       = {Journal of Symbolic Computation},
+  volume        = {42},
+  number        = {3},
+  pages         = {298--312},
+  year          = {2007},
+  doi           = {10.1016/j.jsc.2006.09.004}
+}
+
 @Article{FJR17,
   author        = {Fan, Huijun and Jarvis, Tyler and Ruan, Yongbin},
   title         = {A mathematical theory of the gauged linear sigma model},
@@ -923,6 +979,18 @@ @Article{FJR17
   doi           = {10.2140/gt.2018.22.235}
 }
 
+@Article{FJT07,
+  author        = {Fukuda, Komei and Jensen, Anders N. and Thomas, Rekha R.},
+  title         = {Computing Groebner Fans},
+  journal       = {Math. Comp.},
+  fjournal      = {Mathematics of Computation},
+  volume        = {76},
+  number        = {260},
+  pages         = {2189--2213},
+  year          = {2007},
+  doi           = {10.1090/S0025-5718-07-01986-2}
+}
+
 @InProceedings{FLINT,
   bibkey        = {FLINT},
   author        = {W. B. Hart},
@@ -2171,6 +2239,28 @@ @Article{Tay87
   primaryclass  = {math.GR}
 }
 
+@Article{Tra00,
+  author        = {Tran, Quoc-Nam},
+  title         = {A Fast Algorithm for Gröbner Basis Conversion and its Applications},
+  journal       = {Journal of Symbolic Computation},
+  volume        = {30},
+  number        = {4},
+  pages         = {451--467},
+  year          = {2000},
+  doi           = {10.1006/jsco.1999.0416}
+}
+
+@Article{Tra04,
+  author        = {Tran, Quoc-Nam},
+  title         = {Efficient Groebner walk conversion for implicitization of geometric objects},
+  journal       = {Computer Aided Geometric Design},
+  volume        = {21},
+  number        = {9},
+  pages         = {837--857},
+  year          = {2004},
+  doi           = {10.1016/j.cagd.2004.07.001}
+}
+
 @Misc{VE22,
   author        = {Vaughan-Lee, Michael and Eick, Bettina},
   title         = {SglPPow, Database of groups of prime-power order for some prime-powers, Version 2.3},

experimental/GroebnerWalk/README.md

@@ -0,0 +1,49 @@
+# Gröbner walk
+
+[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.11065978.svg)](https://doi.org/10.5281/zenodo.11065978)
+
+`GroebnerWalk` provides implementations of Gröbner walk algorithms
+for computing Gröbner bases over fields on top of Oscar.jl.
+
+## Usage
+
+This module provides the function `groebner_walk` as interface to the algorithms.
+The following example demonstrates the usage. First, we define the ideal Oscar.jl.
+```julia
+using Oscar
+
+R, (x,y) = QQ[:x, :y]                  # define ring ...
+I = ideal([y^4+ x^3-x^2+x,x^4])        # ... and ideal
+```
+Then, we can pass the ideal to `groebner_walk` to calculate the Gröbner basis.
+
+By default, `groebner_walk` starts with a Gröbner basis with respect to the default ordering on `R`
+and converts this into a Gröbner basis with respect to the lexicographic ordering on `R`.
+This is what the following code block accomplishes.
+```julia
+using Oscar
+
+groebner_walk(I) # compute the Groebner basis
+```
+If one wants to specify `target` and `start` orderings explicitly, above function call needs to be written as follows.
+```julia
+groebner_walk(I, lex(R), default_ordering(R)) # compute the Groebner basis
+```
+
+Additionally, there are certain special ideals provided that are used for benchmarking 
+of this module.
+
+## Status
+At the moment, the standard walk by Collart, Kalkbrener and Mall (1997) and the generic walk by Fukuda et al. (2007) are implemented.
+
+## Contacts
+The library is maintained by Kamillo Ferry (kafe (at) kafe (dot) dev) and Francesco Nowell (francesconowell (at) gmail (dot) com).
+
+## Acknowledgement
+The current implementation is based on an implementation by Jordi Welp. We thank him for 
+laying the groundwork for this package.
+
+## References
+- Collart, S., M. Kalkbrener, and D. Mall. ‘Converting Bases with the Gröbner Walk’. Journal of Symbolic Computation 24, no. 3–4 (September 1997): 465–69. https://doi.org/10.1006/jsco.1996.0145.
+- Fukuda, K., A. N. Jensen, N. Lauritzen, and R. Thomas. ‘The Generic Gröbner Walk’. Journal of Symbolic Computation 42, no. 3 (1 March 2007): 298–312. https://doi.org/10.1016/j.jsc.2006.09.004.
+

experimental/GroebnerWalk/benchmark/agk.jl

@@ -0,0 +1,17 @@
+using Oscar
+
+function agk4(k::Field)
+    R, (x,y,z,u,v) = polynomial_ring(k, ["x","y","z","u","v"])
+
+    o1 = weight_ordering([1,1,1,0,0], degrevlex(R))
+    o2 = weight_ordering([0,0,0,1,1], degrevlex(R))
+
+    F = [
+        u + u^2 - 2*v - 2*u^2*v + 2*u*v^2 - x,
+        -6*u + 2*v + v^2 - 5*v^3 + 2*u*v^2 - 4*u^2*v^2 - y,
+        -2 + 2*u^2 + 6*v - 3*u^2*v^2 - z
+    ]
+
+    return ideal(F), o2, o1
+end
+

experimental/GroebnerWalk/benchmark/benchmarks.jl

@@ -0,0 +1,65 @@
+using Oscar
+
+include("cyclic.jl")
+include("katsura.jl")
+include("agk.jl")
+include("tran3.3.jl")
+
+function benchmark(
+  io,
+  name::String,
+  I::Ideal,
+  target::MonomialOrdering,
+  start::MonomialOrdering
+)
+  print(io, name, ","); flush(io)
+  t = @elapsed groebner_walk($I, $target, $start; algorithm=:standard) 
+  print(io, t, ","); flush(io)
+  t = @elapsed groebner_walk($I, $target, $start; algorithm=:generic)
+  print(io, t, ","); flush(io)
+  t = @elapsed groebner_walk($I, $target, $start; algorithm=:perturbed)
+  print(io, t, ","); flush(io)
+  t = @elapsed groebner_basis($I; ordering=$target)
+  println(io, t); flush(io)
+end
+
+function print_header(io)
+  print(io, "name,standard_walk,generic_walk,perturbed_walk,buchberger\n")
+end
+
+p = 11863279
+Fp = GF(p)
+open("results.csv", "a") do io
+  R, (x, y) = QQ[:x, :y]
+  I = ideal([y^4 + x^3 - x^2 + x, x^4])
+  benchmark(io, "simple", I, lex(R), default_ordering(R))
+  
+  benchmark(io, "cyclic5-QQ", cyclic5(QQ)...)
+  benchmark(io, "cyclic5-Fp", cyclic5(Fp)...)
+
+  benchmark(io, "cyclic6-QQ", cyclic6(QQ)...)
+  benchmark(io, "cyclic6-Fp", cyclic6(Fp)...)
+  
+  I = katsura(6)
+  R = base_ring(I)
+  benchmark(io, "katsura6-QQ", I, lex(R), default_ordering(R))
+
+  Ip = map(gens(I)) do f
+    change_coefficient_ring(Fp, f)
+  end |> ideal
+  Rp = base_ring(Ip)
+  benchmark(io, "katsura6-Fp", Ip, lex(Rp), default_ordering(Rp))
+
+  benchmark(io, "cyclic7-QQ", cyclic7(QQ)...)
+  benchmark(io, "cyclic7-Fp", cyclic7(Fp)...)
+
+  benchmark(io, "agk4-QQ", agk4(QQ)...)
+  benchmark(io, "agk4-Fp", agk4(Fp)...)
+
+  benchmark(io, "tran3.3-QQ", tran33(QQ)...)
+  benchmark(io, "tran3.3-Fp", tran33(Fp)...)
+
+  benchmark(io, "newellp1-QQ", Oscar.newell_patch_with_orderings(QQ)...)
+  benchmark(io, "newellp1-Fp", Oscar.newell_patch_with_orderings(Fp)...)
+end
+

experimental/GroebnerWalk/benchmark/cyclic.jl

@@ -0,0 +1,79 @@
+using Oscar
+
+function cyclic5(k::Field=QQ)
+    R, (a,b,c,d,x) = k[:a,:b,:c,:d,:x]
+    F = [
+        a + b + c + d + x,
+        a*b + b*c + c*d + d*x + x*a,
+        a*b*c + b*c*d + c*d*x + d*x*a + x*a*b,
+        a*b*c*d + b*c*d*x + c*d*x*a + d*x*a*b + x*a*b*c,
+        a*b*c*d*x - 1
+    ]
+
+    return ideal(F), lex(R), default_ordering(R)
+end
+
+function cyclic6(k::Field=QQ)
+    R, (z0,z1,z2,z3,z4,z5) = k[:z0,:z1,:z2,:z3,:z4,:z5]
+    F = [
+      z0 + z1 + z2 + z3 + z4 + z5,
+      z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0,
+      z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1,
+      z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 
+      + z5*z0*z1*z2,
+      z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z0 + z3*z4*z5*z0*z1 
+      + z4*z5*z0*z1*z2 + z5*z0*z1*z2*z3,
+      z0*z1*z2*z3*z4*z5 - 1
+    ]
+
+    return ideal(F), lex(R), default_ordering(R)
+end
+
+function cyclic7(k::Field=QQ)
+    R, (z0, z1, z2, z3, z4, z5, z6) = QQ[:z0, :z1, :z2, :z3, :z4, :z5, :z6]
+    F = [
+        z0 + z1 + z2 + z3 + z4 + z5 + z6,
+        z0 * z1 + z1 * z2 + z2 * z3 + z3 * z4 + z4 * z5 + z5 * z6 + z6 * z0,
+        z0 * z1 * z2 + z1 * z2 * z3 + z2 * z3 * z4 + z3 * z4 * z5 + z4 * z5 * z6 + z5 * z6 * z0 + z6 * z0 * z1,
+        z0 * z1 * z2 * z3 + z1 * z2 * z3 * z4 + z2 * z3 * z4 * z5 + z3 * z4 * z5 * z6 + z4 * z5 * z6 * z0
+        + z5 * z6 * z0 * z1 + z6 * z0 * z1 * z2,
+        z0 * z1 * z2 * z3 * z4 + z1 * z2 * z3 * z4 * z5 + z2 * z3 * z4 * z5 * z6 + z3 * z4 * z5 * z6 * z0
+        + z4 * z5 * z6 * z0 * z1 + z5 * z6 * z0 * z1 * z2 + z6 * z0 * z1 * z2 * z3,
+        z0 * z1 * z2 * z3 * z4 * z5 + z1 * z2 * z3 * z4 * z5 * z6 + z2 * z3 * z4 * z5 * z6 * z0 + z3 * z4 * z5 * z6 * z0 * z1
+        + z4 * z5 * z6 * z0 * z1 * z2 + z5 * z6 * z0 * z1 * z2 * z3 + z6 * z0 * z1 * z2 * z3 * z4,
+        z0 * z1 * z2 * z3 * z4 * z5 * z6 - 1
+    ]
+
+    return ideal(F), lex(R), default_ordering(R)
+end
+
+function cyclic8(k::Field=QQ)
+    R, (z0, z1, z2, z3, z4, z5, z6, z7) = k[:z0, :z1, :z2, :z3, :z4, :z5, :z6, :z7]
+
+    F = [
+        z0 + z1 + z2 + z3 + z4 + z5 + z6 + z7,
+
+        z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z6 + z6*z7 + z7*z0,
+
+        z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z6 + z5*z6*z7
+        + z6*z7*z0 + z7*z0*z1,
+
+        z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z6 + z4*z5*z6*z7
+        + z5*z6*z7*z0 + z6*z7*z0*z1 + z7*z0*z1*z2,
+
+        z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z6 + z3*z4*z5*z6*z7
+        + z4*z5*z6*z7*z0 + z5*z6*z7*z0*z1 + z6*z7*z0*z1*z2 + z7*z0*z1*z2*z3,
+
+        z0*z1*z2*z3*z4*z5 + z1*z2*z3*z4*z5*z6 + z2*z3*z4*z5*z6*z7 + z3*z4*z5*z6*z7*z0
+        + z4*z5*z6*z7*z0*z1 + z5*z6*z7*z0*z1*z2 + z6*z7*z0*z1*z2*z3 + z7*z0*z1*z2*z3*z4,
+    
+        z0*z1*z2*z3*z4*z5*z6 + z1*z2*z3*z4*z5*z6*z7 + z2*z3*z4*z5*z6*z7*z0
+        + z3*z4*z5*z6*z7*z0*z1 + z4*z5*z6*z7*z0*z1*z2 + z5*z6*z7*z0*z1*z2*z3
+        + z6*z7*z0*z1*z2*z3*z4 + z7*z0*z1*z2*z3*z4*z5,
+
+        z0*z1*z2*z3*z4*z5*z6*z7 - 1
+    ]
+
+    return ideal(F), lex(R), default_ordering(R)
+end
+

experimental/GroebnerWalk/benchmark/tran3.3.jl

@@ -0,0 +1,10 @@
+using Oscar 
+
+function tran33(k::Field)
+    R, (x,y,z) = k[:x,:y,:z]
+
+    F = [16 + 3*x^3+16*x^2*z+14*x^2*y^3, 6+y^3*z+17*x^2*z^2+7*x*y^2*z^2+13*x^3*z^2]
+
+    return ideal(F), lex(R), default_ordering(R)
+end
+

experimental/GroebnerWalk/docs/doc.main

@@ -0,0 +1,6 @@
+[
+  "Gröbner walk" => [
+    "introduction.md",
+    "special-ideals.md"
+  ]
+]

experimental/GroebnerWalk/docs/src/introduction.md

@@ -0,0 +1,22 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Usage
+
+The Gröbner walk is an approach to reduce the computational complexity of Gröbner basis computations as proposed by [AGK97](@cite).
+These incarnations of the Gröbner walk refer to a family of algorithms that perform a reverse local search on the cones of the Gröbner fan.
+Then, a Gröbner basis is calculated for each encountered cone while reusing the generators obtained from the previous cone.
+
+The implemented algorithms may be accessed using the following function.
+
+```@docs
+    groebner_walk(
+      I::MPolyIdeal, 
+      target::MonomialOrdering = lex(base_ring(I)),
+      start::MonomialOrdering = default_ordering(base_ring(I));
+      perturbation_degree = ngens(base_ring(I)),
+      algorithm::Symbol = :standard
+    )
+```
+    

experimental/GroebnerWalk/docs/src/special-ideals.md

@@ -0,0 +1,16 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Special ideals used for benchmarking
+
+We bundle a couple of special ideals useful for benchmarking of the Gröbner walk.
+
+```@docs
+    newell_patch(k::Union{QQField, QQBarFieldElem}, n::Int=1)
+    newell_patch(k::Field, n::Int=1)
+```
+
+```@docs
+    newell_patch_with_orderings(k::Field, n::Int=1)
+```