experimental: Gröbner Walk #3821
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| # Gröbner walk | ||
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| [](https://doi.org/10.5281/zenodo.11065978) | ||
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| `GroebnerWalk` provides implementations of Gröbner walk algorithms | ||
| for computing Gröbner bases over fields on top of Oscar.jl. | ||
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| ## Usage | ||
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| 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 | ||
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| 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. | ||
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| 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 | ||
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| 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 | ||
| ``` | ||
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| Additionally, there are certain special ideals provided that are used for benchmarking | ||
| of this module. | ||
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| ## Status | ||
| At the moment, the standard walk by Collart, Kalkbrener and Mall (1997) and the generic walk by Fukuda et al. (2007) are implemented. | ||
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| ## Contacts | ||
| The library is maintained by Kamillo Ferry (kafe (at) kafe (dot) dev) and Francesco Nowell (francesconowell (at) gmail (dot) com). | ||
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| ## Acknowledgement | ||
| The current implementation is based on an implementation by Jordi Welp. We thank him for | ||
| laying the groundwork for this package. | ||
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| ## 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. | ||
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| using Oscar | ||
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| function agk4(k::Field) | ||
| R, (x,y,z,u,v) = polynomial_ring(k, ["x","y","z","u","v"]) | ||
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| o1 = weight_ordering([1,1,1,0,0], degrevlex(R)) | ||
| o2 = weight_ordering([0,0,0,1,1], degrevlex(R)) | ||
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| 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 | ||
| ] | ||
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| return ideal(F), o2, o1 | ||
| end | ||
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| using Oscar | ||
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| include("cyclic.jl") | ||
| include("katsura.jl") | ||
| include("agk.jl") | ||
| include("tran3.3.jl") | ||
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| 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 | ||
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| function print_header(io) | ||
| print(io, "name,standard_walk,generic_walk,perturbed_walk,buchberger\n") | ||
| end | ||
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| 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)) | ||
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| benchmark(io, "cyclic5-QQ", cyclic5(QQ)...) | ||
| benchmark(io, "cyclic5-Fp", cyclic5(Fp)...) | ||
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| benchmark(io, "cyclic6-QQ", cyclic6(QQ)...) | ||
| benchmark(io, "cyclic6-Fp", cyclic6(Fp)...) | ||
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| I = katsura(6) | ||
| R = base_ring(I) | ||
| benchmark(io, "katsura6-QQ", I, lex(R), default_ordering(R)) | ||
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| 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)) | ||
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| benchmark(io, "cyclic7-QQ", cyclic7(QQ)...) | ||
| benchmark(io, "cyclic7-Fp", cyclic7(Fp)...) | ||
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| benchmark(io, "agk4-QQ", agk4(QQ)...) | ||
| benchmark(io, "agk4-Fp", agk4(Fp)...) | ||
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| benchmark(io, "tran3.3-QQ", tran33(QQ)...) | ||
| benchmark(io, "tran3.3-Fp", tran33(Fp)...) | ||
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| benchmark(io, "newellp1-QQ", Oscar.newell_patch_with_orderings(QQ)...) | ||
| benchmark(io, "newellp1-Fp", Oscar.newell_patch_with_orderings(Fp)...) | ||
| end | ||
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| using Oscar | ||
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| 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 | ||
| ] | ||
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| return ideal(F), lex(R), default_ordering(R) | ||
| end | ||
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| 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 | ||
| ] | ||
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| return ideal(F), lex(R), default_ordering(R) | ||
| end | ||
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| 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 | ||
| ] | ||
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| return ideal(F), lex(R), default_ordering(R) | ||
| end | ||
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| function cyclic8(k::Field=QQ) | ||
| R, (z0, z1, z2, z3, z4, z5, z6, z7) = k[:z0, :z1, :z2, :z3, :z4, :z5, :z6, :z7] | ||
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| F = [ | ||
| z0 + z1 + z2 + z3 + z4 + z5 + z6 + z7, | ||
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| z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z6 + z6*z7 + z7*z0, | ||
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| z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z6 + z5*z6*z7 | ||
| + z6*z7*z0 + z7*z0*z1, | ||
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| 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, | ||
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| 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, | ||
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| 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, | ||
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| 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, | ||
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| z0*z1*z2*z3*z4*z5*z6*z7 - 1 | ||
| ] | ||
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| return ideal(F), lex(R), default_ordering(R) | ||
| end | ||
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| using Oscar | ||
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| function tran33(k::Field) | ||
| R, (x,y,z) = k[:x,:y,:z] | ||
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| 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] | ||
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| return ideal(F), lex(R), default_ordering(R) | ||
| end | ||
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| [ | ||
| "Gröbner walk" => [ | ||
| "introduction.md", | ||
| "special-ideals.md" | ||
| ] | ||
| ] |
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| ```@meta | ||
| CurrentModule = Oscar | ||
| ``` | ||
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| # Usage | ||
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| 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. | ||
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| The implemented algorithms may be accessed using the following function. | ||
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| ```@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 | ||
| ) | ||
| ``` | ||
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| ```@meta | ||
| CurrentModule = Oscar | ||
| ``` | ||
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| # Special ideals used for benchmarking | ||
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| We bundle a couple of special ideals useful for benchmarking of the Gröbner walk. | ||
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| ```@docs | ||
| newell_patch(k::Union{QQField, QQBarFieldElem}, n::Int=1) | ||
| newell_patch(k::Field, n::Int=1) | ||
| ``` | ||
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| ```@docs | ||
| newell_patch_with_orderings(k::Field, n::Int=1) | ||
| ``` |
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Concerning this benchmark and the others below: There is
src/Rings/special_ideals.jlin which someone already has added thekatsura-nexamples. Maybe we can add your benchmarks there and remove these different files, I assume your benchmarks are also of interest to other developers/users.There was a problem hiding this comment.
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I started adding the ideals to a
special-ideals.jlin the experimental module. I will add the others with references, as soon as this is done I will move this tosrc/Rings/special_ideals.jl(in a future PR)