groebner(polynomials; options...)
Computes a Groebner basis of the ideal generated by polynomials
.
polynomials
: an array of polynomials. Currently supports polynomials from AbstractAlgebra.jl, Nemo.jl, and DynamicPolynomials.jl. For AbstractAlgebra.jl and Nemo.jl, the coefficients of polynomials must be either in GF(p)
, in Native.GF(p)
, or in QQ
.
The groebner
routine takes the following optional arguments:
reduced
: If the returned basis must be autoreduced and unique (default is true
).
ordering
: Specifies the monomial ordering. Available monomial orderings are:
InputOrdering()
for inferring the ordering from the given polynomials
(default),
Lex(args...)
for lexicographic,
DegLex(args...)
for degree lexicographic,
DegRevLex(args...)
for degree reverse lexicographic,
WeightedOrdering(args...)
for weighted ordering,
ProductOrdering(args...)
for block ordering,
MatrixOrdering(args...)
for matrix ordering. For details and examples see the corresponding documentation page.
certify
: Certify the obtained basis. When this option is false
, the algorithm is randomized, and the result is correct with high probability. When this option is true
, the result is guaranteed to be correct in case the ideal is homogeneous (default is false
).
linalg
: Linear algebra backend. Available options are:
:auto
for the automatic choice (default),
:deterministic
for deterministic sparse linear algebra,
:randomized
for probabilistic sparse linear algebra.
threaded
: The use of multi-threading. Available options are:
:auto
for the automatic choice (default),
:no
never use multi-threading,
:yes
allow the use of multi-threading.
Additionally, you can set the environment variable GROEBNER_NO_THREADED
to 1
to disable all multi-threading in Groebner.jl. In this case, the environment variable takes precedence over the threaded
option.
monoms
: Monomial representation used in the computations. The algorithm tries to automatically choose the most suitable monomial representation. Otherwise, set monoms
to one of the following:
:auto
for the automatic choice (default),
:dense
for classic dense exponent vectors,
:packed
for packed representation,
:sparse
for sparse representation.
modular
: Modular computation algorithm. Only has effect when computing basis over the rational numbers. Possible options are:
:auto
for the automatic choice (default),
:classic_modular
for the classic multi-modular algorithm,
:learn_and_apply
for the learn & apply algorithm.
seed
: The seed for randomization. Default value is 42
. Groebner uses Random.Xoshiro
and Random.MersenneTwister
for random number generation.
loglevel
: Logging level, one of :all
, :debug
, :info
, :warn
, :no
, or an integer. An integer 0
is equivalent to :info
, and lower integer values mean more verbose. Default is :info
.
maxpairs
: The maximum number of critical pairs used at once in matrix construction in the F4 algorithm. By default, this is not specified. Tweak this option to try to lower the amount of RAM consumed.
homogenize
: Controls the use of homogenization in the algorithm. Possible options are:
:yes
, always homogenize the input ideal,
:no
, never homogenize the input ideal,
:auto
, for the automatic choice (default).
statistics
: After the function exits, print some runtime statistics. Possible options are:
:no
, print nothing (default),
:timings
, print the table with timings and allocations. Note that Groebner.performance_counters_enabled()
must be set to true
for this to have effect.
Using DynamicPolynomials.jl:
using Groebner, DynamicPolynomials
@polyvar x y
groebner([x*y^2 + x, y*x^2 + y])
Using AbstractAlgebra.jl:
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"]
groebner([x*y^2 + x, y*x^2 + y])
Using Nemo.jl:
using Groebner, Nemo
R, (x, y) = GF(2^30+3)["x", "y"]
groebner([x*y^2 + x, y*x^2 + y])
Or, say, in another monomial ordering:
# y > x
groebner([x*y^2 + x, y*x^2 + y], ordering=Lex(y, x))
# degree reverse lexicographic
groebner([x*y^2 + x, y*x^2 + y], ordering=DegRevLex())
The function groebner
is thread-safe.
The default algorithm is probabilistic (with certify=false
). Results are correct with high probability, however, no precise bound on the probability is known.
groebner_with_change_matrix(polynomials; options...)
Computes a Groebner basis of the ideal generated by polynomials
and emits a change matrix, that is, a map from the original generators to basis elements.
polynomials
: an array of polynomials. Currently supports polynomials from AbstractAlgebra.jl, Nemo.jl, and DynamicPolynomials.jl. For AbstractAlgebra.jl and Nemo.jl, the coefficients of polynomials must be either in GF(p)
, in Native.GF(p)
, or in QQ
.
Same as for groebner
.
Using AbstractAlgebra.jl:
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"]
f = [x*y^2 + x, y*x^2 + y]
g, m = groebner_with_change_matrix(f, ordering=DegRevLex())
@assert isgroebner(g, ordering=DegRevLex())
@assert m * f == g
Same as for groebner
.
isgroebner(polynomials; options...)
Checks if polynomials
forms a Groebner basis.
polynomials
: an array of polynomials. Currently supports polynomials from AbstractAlgebra.jl, Nemo.jl, and DynamicPolynomials.jl. For AbstractAlgebra.jl and Nemo.jl, the coefficients of polynomials must be either in GF(p)
, in Native.GF(p)
, or in QQ
.
The isgroebner
routine takes the following optional arguments:
ordering
: Specifies the monomial ordering. Available monomial orderings are:
InputOrdering()
for inferring the ordering from the given polynomials
(default),
Lex()
for lexicographic,
DegLex()
for degree lexicographic,
DegRevLex()
for degree reverse lexicographic,
WeightedOrdering(weights)
for weighted ordering,
ProductOrdering(args...)
for block ordering,
MatrixOrdering(matrix)
for matrix ordering.
For details and examples see the corresponding documentation page.
certify
: Use deterministic algorithm (default is false
).
seed
: The seed for randomization. Default value is 42
. Groebner uses Random.Xoshiro
and Random.MersenneTwister
for random number generation.
loglevel
: Logging level, one of :all
, :debug
, :info
, :warn
, :no
, or an integer. An integer 0
is equivalent to :info
, and lower integer values mean more verbose. Default is :info
.
statistics
: After the function exits, print some runtime statistics. Possible options are:
:no
, do not print anything (default),
:timings
, print the table with timings and allocations. Note that Groebner.performance_counters_enabled()
must be set to true
for this to have effect.
Using DynamicPolynomials
:
using Groebner, DynamicPolynomials
@polyvar x y;
isgroebner([x*y^2 + x, y*x^2 + y])
Using AbstractAlgebra
:
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"]
isgroebner([x*y^2 + x, y*x^2 + y])
The function isgroebner
is thread-safe.
normalform(basis, tobereduced; options...)
Computes the normal form of polynomials tobereduced
with respect to basis
.
basis
: an array of polynomials, a Groebner basis.
tobereduced
: can be either a single polynomial or an array of polynomials.
The normalform
routine takes the following optional arguments:
check
: Check if the given array basis
forms a Groebner basis (default is false
).
ordering
: Specifies the monomial ordering. Available monomial orderings are:
InputOrdering()
for inferring the ordering from the given polynomials
(default),
Lex()
for lexicographic,
DegLex()
for degree lexicographic,
DegRevLex()
for degree reverse lexicographic,
WeightedOrdering(weights)
for weighted ordering,
ProductOrdering(args...)
for block ordering,
MatrixOrdering(matrix)
for matrix ordering.
For details and examples see the corresponding documentation page.
loglevel
: Logging level, one of :all
, :debug
, :info
, :warn
, :no
, or an integer. An integer 0
is equivalent to :info
, and lower integer values mean more verbose. Default is :info
.
statistics
: After the function exits, print some runtime statistics. Possible options are:
:no
, do not print anything (default),
:timings
, print the table with timings and allocations. Note that Groebner.performance_counters_enabled()
must be set to true
for this to have effect.
Reducing a single polynomial:
using Groebner, DynamicPolynomials
@polyvar x y;
normalform([y^2 + x, x^2 + y], x^2 + y^2 + 1)
Or, reducing two polynomials at a time:
using Groebner, DynamicPolynomials
@polyvar x y;
normalform([y^2 + x, x^2 + y], [x^2 + y^2 + 1, x^10*y^10])
The function normalform
is thread-safe.
kbase(basis; options...)
Returns the standard basis of the quotient space of the ideal generated by basis
.
basis
: an array of polynomials, a Groebner basis.
The kbase
routine takes the following optional arguments:
check
: Check if the given array basis
forms a Groebner basis (default is false
).
loglevel
: Logging level, one of :all
, :debug
, :info
, :warn
, :no
, or an integer. An integer 0
is equivalent to :info
, and lower integer values mean more verbose. Default is :info
.
statistics
: After the function exits, print some runtime statistics. Possible options are:
:no
, do not print anything (default),
:timings
, print the table with timings and allocations. Note that Groebner.performance_counters_enabled()
must be set to true
for this to have effect.
using Groebner, DynamicPolynomials
@polyvar x y;
kbase([y^2 + x, x^2 + y])
The function kbase
is thread-safe.
A list of all monomial orderings supported by Groebner.jl. An ordering can be set by passing it with the keyword argument ordering
. See below for some examples.
Lex()
Lex(variables)
Lex(variables...)
Lexicographical monomial ordering.
We use the definition from Chapter 1, Computational Commutative Algebra 1, by Martin Kreuzer, Lorenzo Robbiano.
DOI: https://doi.org/10.1007/978-3-540-70628-1.
Dura Lex, sed Lex.
compile
: If true
, compiles and uses a specialized monomial comparator function, which may speed up the runtime (default is false
). This option is experimental.
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"];
# Lexicographical ordering with x > y
groebner([x*y + x, x + y^2], ordering=Lex())
# Lexicographical ordering with y > x
groebner([x*y + x, x + y^2], ordering=Lex([y, x]))
# Lexicographical ordering with x > y
# Note that both syntax -- Lex([...]) and Lex(...) -- are allowed
groebner([x*y + x, x + y^2], ordering=Lex(x, y))
DegLex()
DegLex(variables)
DegLex(variables...)
Degree lexicographical monomial ordering.
We use the definition from Chapter 1, Computational Commutative Algebra 1, by Martin Kreuzer, Lorenzo Robbiano.
DOI: https://doi.org/10.1007/978-3-540-70628-1.
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"];
# Degree lexicographical ordering with x > y
groebner([x*y + x, x + y^2], ordering=DegLex())
# Degree lexicographical ordering with y > x
groebner([x*y + x, x + y^2], ordering=DegLex([y, x]))
DegRevLex()
DegRevLex(variables)
DegRevLex(variables...)
Degree reverse lexicographical monomial ordering.
We use the definition from Chapter 1, Computational Commutative Algebra 1, by Martin Kreuzer, Lorenzo Robbiano.
DOI: https://doi.org/10.1007/978-3-540-70628-1.
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"];
# Degree reverse lexicographical ordering with x > y
groebner([x*y + x, x + y^2], ordering=DegRevLex())
# Degree reverse lexicographical ordering with y > x
groebner([x*y + x, x + y^2], ordering=DegRevLex(y, x))
InputOrdering()
Preserves the monomial ordering defined on the input polynomials.
This is the default value for the ordering
keyword argument in the exported functions.
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"]
# Uses the ordering `InputOrdering`, which, in this case,
# defaults to the lexicographical ordering with x > y
groebner([x*y + x, x + y^2])
WeightedOrdering(weights)
Weighted monomial ordering.
Only positive weights are supported.
using Groebner, AbstractAlgebra
R, (x, y) = QQ["x", "y"];
# x has weight 3, y has weight 1
ord = WeightedOrdering([3, 1])
groebner([x*y + x, x + y^2], ordering=ord)
ProductOrdering(ord1, ord2)
Product monomial ordering. Compares monomials by ord1
, then breaks ties by ord2
.
Can also be constructed with *
.
using Groebner, AbstractAlgebra
R, (x, y, z, w) = QQ["x", "y", "z", "w"];
# Ordering with x, y > w, z
ord = ProductOrdering(DegRevLex(x, y), DegRevLex(w, z))
groebner([x*y + w, y*z - w], ordering=ord)
You can also use the *
operator:
using Groebner, AbstractAlgebra
R, (x, y, z, t) = QQ["x", "y", "z", "t"];
ord1 = Lex(t)
ord2 = DegRevLex(x, y, z)
# t >> x, y, z
ord = ord1 * ord2
groebner([x*y*z + z, t * z - 1], ordering=ord)
MatrixOrdering(matrix)
MatrixOrdering(Vector{Vector})
Matrix monomial ordering.
using Groebner, AbstractAlgebra
R, (x, y, z, w) = QQ["x", "y", "z", "w"];
# the number of columns equal to the number of variables
ord = MatrixOrdering([
1 0 0 2;
0 0 1 2;
0 1 1 1;
])
groebner([x*y + w, y*z - w], ordering=ord)
groebner_learn(polynomials; options...)
Computes a Groebner basis of the ideal generated by polynomials
and emits the trace.
The trace can be used to speed up the computation of other Groebner bases, which should be specializations of the same ideal as the one groebner_learn
has been applied to.
See also groebner_apply!
.
polynomials
: an array of polynomials. Must be polynomials from AbstractAlgebra.jl over GF(p)
.
Using groebner_learn
and groebner_apply!
over the same ground field:
using Groebner, AbstractAlgebra
R, (x, y) = GF(2^31-1)["x", "y"]
# Learn
trace, gb_1 = groebner_learn([x*y^2 + x, y*x^2 + y])
# Apply (same support, different coefficients)
flag, gb_2 = groebner_apply!(trace, [2x*y^2 + 3x, 4y*x^2 + 5y])
@assert flag
Using groebner_learn
and groebner_apply!
over different ground fields:
using Groebner, AbstractAlgebra
R, (x, y) = GF(2^31-1)["x", "y"]
# Learn
trace, gb_1 = groebner_learn([x*y^2 + x, y*x^2 + y], ordering=DegRevLex())
# Create a ring with a different modulo
R2, (x2, y2) = GF(2^30+3)["x", "y"]
# Apply (different modulo)
flag, gb_2 = groebner_apply!(
trace,
[2x2*y2^2 + 3x2, 4y2*x2^2 + 5y2],
ordering=DegRevLex()
)
@assert flag
@assert gb_2 == groebner([2x2*y2^2 + 3x2, 4y2*x2^2 + 5y2], ordering=DegRevLex())
Using groebner_apply!
in batches (works only in :degrevlex
at the moment):
using Groebner, AbstractAlgebra
R, (x, y) = polynomial_ring(GF(2^31-1), ["x", "y"], internal_ordering=:degrevlex)
# Learn
trace, gb_1 = groebner_learn([x*y^2 + x, y*x^2 + y])
# Create rings with some other moduli
R2, (x2, y2) = polynomial_ring(GF(2^30+3), ["x", "y"], internal_ordering=:degrevlex)
R3, (x3, y3) = polynomial_ring(GF(2^27+29), ["x", "y"], internal_ordering=:degrevlex)
# Two specializations of the same ideal
batch = ([2x2*y2^2 + 3x2, 4y2*x2^2 + 5y2], [4x3*y3^2 + 4x3, 5y3*x3^2 + 7y3])
# Apply for two sets of polynomials at once
flag, (gb_2, gb_3) = groebner_apply!(trace, batch)
@assert flag
@assert (gb_2, gb_3) == map(groebner, batch)
Perhaps, in a more involved example, we will compute Groebner bases of the Katsura-9 system:
using Groebner, AbstractAlgebra, BenchmarkTools
# Create the system
kat = Groebner.katsuran(9, k=ZZ, internal_ordering=:degrevlex)
# Reduce the coefficients modulo 5 different primes
kat_0 = map(f -> map_coefficients(c -> GF(2^30 + 3)(c), f), kat)
kat_1 = map(f -> map_coefficients(c -> GF(2^30 + 7)(c), f), kat)
kat_2 = map(f -> map_coefficients(c -> GF(2^30 + 9)(c), f), kat)
kat_3 = map(f -> map_coefficients(c -> GF(2^30 + 15)(c), f), kat)
kat_4 = map(f -> map_coefficients(c -> GF(2^30 + 19)(c), f), kat)
# Learn the trace
trace, gb_0 = groebner_learn(kat_0);
# Compare the performance of applying with 1 input and with 4 different inputs:
# Apply for one system
@btime groebner_apply!($trace, $kat_1);
# 46.824 ms (19260 allocations: 24.48 MiB)
# Apply for a batch of four systems
@btime groebner_apply!($trace, $(kat_1, kat_2, kat_3, kat_4));
# 72.813 ms (23722 allocations: 59.44 MiB)
Observe the better amortized performance of the batched groebner_apply!
.
The function groebner_learn
is thread-safe.
groebner_apply!(trace, polynomials; options...)
groebner_apply!(trace, batch::NTuple{N, Vector}; options...)
Computes a Groebner basis of the ideal generated by polynomials
following the given trace
.
See also groebner_learn
.
trace
: a trace from groebner_learn
.
polynomials
: an array of polynomials. Must be polynomials from AbstractAlgebra.jl over GF(p)
. It is possible to supply a tuple of N
arrays to compute N
Groebner bases simultaneously, which could be more efficient overall than computing them in separate.
The groebner_apply!
routine automatically inherits most parameters from the given trace
.
For examples, see the documentation of groebner_learn
.
This function is not thread-safe, since it mutates the trace
.