Algebraic Systems Solving

rational_parametrization(I::Ideal{T} where T <: MPolyRingElem, <keyword arguments>)

Given an ideal I with a finite solution set over the complex numbers, return the rational parametrization of the ideal with a given precision (default 32 bits).

Note: At the moment only QQ is supported as ground field. If the dimension of the ideal is greater than zero an ErrorException is thrown.

Arguments

• I::Ideal{T} where T <: MPolyRingElem: input generators.
• initial_hts::Int=17: initial hash table size log_2.
• nr_thrds::Int=1: number of threads for parallel linear algebra.
• max_nr_pairs::Int=0: maximal number of pairs per matrix, only bounded by minimal degree if 0.
• la_option::Int=2: linear algebra option: exact sparse-dense (1), exact sparse (2, default), probabilistic sparse-dense (42), probabilistic sparse(44).
• info_level::Int=0: info level printout: off (0, default), summary (1), detailed (2).
• precision::Int=32: bit precision for the computed solutions.

Examples

julia> using AlgebraicSolving

julia> R,(x1,x2,x3) = polynomial_ring(QQ, ["x1","x2","x3"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x1, x2, x3])

julia> I = Ideal([x1+2*x2+2*x3-1, x1^2+2*x2^2+2*x3^2-x1, 2*x1*x2+2*x2*x3-x2])
QQMPolyRingElem[x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2, 2*x1*x2 + 2*x2*x3 - x2]

julia> rational_parametrization(I)
AlgebraicSolving.RationalParametrization([:x1, :x2, :x3], ZZRingElem[], 84*x^4 - 40*x^3 + x^2 + x, 336*x^3 - 120*x^2 + 2*x + 1, Nemo.QQPolyRingElem[184*x^3 - 80*x^2 + 4*x + 1, 36*x^3 - 18*x^2 + 2*x])

rational_parametrization(I::ParametricIdeal{K}; <keyword arguments>) -> (Vector{Symbol}, Vector{MPoly{K}}, MPoly{K}, MPoly{K}, Vector{MPoly{K}})

Computes the parametric rational parametrization associated with the parametric ideal I. The rational parametrization is computed by evaluating the ideal at a number of parameter values and interpolating the coefficients of the resulting rational parametrizations.

Arguments

• retry::Int=10: the maximum number of consecutive failures allowed when computing the rational parametrization or interpolating the coefficients.
• nr_thrds::Int=1: the number of threads to use for parallel computations.
• worker_pool::AbstractWorkerPool=default_worker_pool(): the worker pool to use for parallel computations.
• show_progress::Bool=false: whether to show a progress bar while computing the rational parametrization.

real_solutions(I::Ideal{T} where T <: MPolyRingElem, <keyword arguments>)

Given an ideal I with a finite solution set over the complex numbers, return the real roots of the ideal with a given precision (default 32 bits).

Note: At the moment only QQ is supported as ground field. If the dimension of the ideal is greater than zero an ErrorException is thrown.

Arguments

• I::Ideal{T} where T <: MPolyRingElem: input generators.
• initial_hts::Int=17: initial hash table size log_2.
• nr_thrds::Int=1: number of threads for parallel linear algebra.
• max_nr_pairs::Int=0: maximal number of pairs per matrix, only bounded by minimal degree if 0.
• la_option::Int=2: linear algebra option: exact sparse-dense (1), exact sparse (2, default), probabilistic sparse-dense (42), probabilistic sparse(44).
• info_level::Int=0: info level printout: off (0, default), summary (1), detailed (2).
• precision::Int=32: bit precision for the computed solutions.
• interval::Bool=false: return real solutions as intervals.

Examples

julia> using AlgebraicSolving

julia> R,(x1,x2,x3) = polynomial_ring(QQ, ["x1","x2","x3"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x1, x2, x3])

julia> I = Ideal([x1+2*x2+2*x3-1, x1^2+2*x2^2+2*x3^2-x1, 2*x1*x2+2*x2*x3-x2])
QQMPolyRingElem[x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2, 2*x1*x2 + 2*x2*x3 - x2]

julia> real_solutions(I)
4-element Vector{Vector{QQFieldElem}}:
 [1454069069139182487//2305843009213693952, 1454069069139182487//4611686018427387904, -154187553040555781639//1180591620717411303424]
 [1, 0, 0]
 [8357884734418025835//36893488147419103232, 4178942367209012917//36893488147419103232, 40355437357166103123//147573952589676412928]
 [6148914691236517205//18446744073709551616, 123527304065019319//85070591730234615865843651857942052864, 196765270119568550571//590295810358705651712]

julia> real_solutions(I, interval=true)
4-element Vector{Vector{Vector{QQFieldElem}}}:
 [[11632552553113459895//18446744073709551616, 11632552553113459897//18446744073709551616], [5816276276556729947//18446744073709551616, 5816276276556729949//18446744073709551616], [-19273444130069472705//147573952589676412928, -77093776520277890819//590295810358705651712]]
 [[1, 1], [0, 0], [0, 0]]
 [[2089471183604506445//9223372036854775808, 4178942367209012945//18446744073709551616], [1044735591802253225//9223372036854775808, 2089471183604506467//18446744073709551616], [20177718678583051561//73786976294838206464, 10088859339291525781//36893488147419103232]]
 [[6148914691236517197//18446744073709551616, 6148914691236517213//18446744073709551616], [-9388075108941468237//170141183460469231731687303715884105728, 9882184325201545513//170141183460469231731687303715884105728], [98382635059784275285//295147905179352825856, 49191317529892137643//147573952589676412928]]

rational_solutions(I::Ideal{T} where T <: MPolyRingElem, <keyword arguments>)

Given an ideal I with a finite solution set over the complex numbers, return the rational roots of the ideal.

Arguments

• I::Ideal{T} where T <: MPolyRingElem: input generators.
• initial_hts::Int=17: initial hash table size log_2.
• nr_thrds::Int=1: number of threads for parallel linear algebra.
• max_nr_pairs::Int=0: maximal number of pairs per matrix, only bounded by minimal degree if 0.
• la_option::Int=2: linear algebra option: exact sparse-dense (1), exact sparse (2, default), probabilistic sparse-dense (42), probabilistic sparse(44).
• info_level::Int=0: info level printout: off (0, default), summary (1), detailed (2).
• precision::Int=32: bit precision for the computed solutions.

Examples

julia> using AlgebraicSolving

julia> R,(x1,x2,x3) = polynomial_ring(QQ, ["x1","x2","x3"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x1, x2, x3])

julia> I = Ideal([x1+2*x2+2*x3-1, x1^2+2*x2^2+2*x3^2-x1, 2*x1*x2+2*x2*x3-x2])
QQMPolyRingElem[x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2, 2*x1*x2 + 2*x2*x3 - x2]

julia> rat_sols = rational_solutions(I)
2-element Vector{Vector{QQFieldElem}}:
 [1//3, 0, 1//3]
 [1, 0, 0]

julia> map(r->map(p->evaluate(p, r), I.gens), rat_sols)
2-element Vector{Vector{QQFieldElem}}:
 [0, 0, 0]
 [0, 0, 0]

curve_rational_parametrization(I::Ideal{<:MPolyRingElem}, <keyword arguments>)

Given a radical ideal I with solution set X being of dimension 1 over the complex numbers, return a rational curve parametrization of the one-dimensional irreducible components of X.

Important: In the output, the variables x and y correspond respectively to the last and second-to-last entries of the vars attribute.

Note: At the moment only QQ is supported as ground field. If the dimension of the ideal is not one an ErrorException is thrown.

Arguments

• I::Ideal{<:QQMPolyRingElem}: input generators.
• info_level::Int=0: info level printout: off (0, default), summary (1), detailed (2).
• cfs_lfs::Vector{Vector{ZZRingElem}} = []: coefficients for the above linear forms
• nr_thrds::Int=1: number of threads for msolve

Examples

julia> using AlgebraicSolving

julia> R, (x1,x2,x3) = polynomial_ring(QQ, ["x1","x2","x3"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x1, x2, x3])

julia> I = Ideal([x1+2*x2+2*x3-1, x1^2+2*x2^2+2*x3^2-x1])
QQMPolyRingElem[x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2]

julia> curve_rational_parametrization(I)
AlgebraicSolving.CurveRationalParametrization([:x1, :x2, :x3, :_Z2, :_Z1], Vector{ZZRingElem}[[0, 0, 1, 0, -1], [0, 1, 0, -1, 0]], x^2 + 4//3*x*y - 1//3*x + y^2 - 1//3*y, 4//3*x + 2*y - 1//3, QQMPolyRingElem[4//3*x^2 - 4//3*x*y + 2//3*x + 4//3*y - 1//3, -2*x^2 - 4//3*x*y + 2//3*x + 1//3*y, 4//3*x^2 + 2*x*y - 1//3*x])

julia> curve_rational_parametrization(I, cfs_lfs=[[-3,2,2], [1,4,-3]])
AlgebraicSolving.CurveRationalParametrization([:x1, :x2, :x3, :_Z2, :_Z1], Vector{ZZRingElem}[[-3, 2, 2, 0, -1], [1, 4, -3, -1, 0]], 127//128*x^2 + 3//8*x*y + 161//64*x + y^2 - 7//8*y - 49//128, 3//8*x + 2*y - 7//8, QQMPolyRingElem[-3//32*x^2 - 1//2*x*y + 5//16*x + 1//2*y - 7//32, -1//4*x^2 + 1//8*x*y - 3//4*x + 3//8*y, 19//64*x^2 + 1//8*x*y + 25//32*x + 3//8*y - 21//64])

julia> curve_rational_parametrization(I, cfs_lfs=[[-3,2,2,-1,-2], [1,4,-3,2,-1]])
AlgebraicSolving.CurveRationalParametrization([:x1, :x2, :x3, :_Z2, :_Z1], Vector{ZZRingElem}[[-3, 2, 2, -1, -2], [1, 4, -3, 2, -1]], 244//181*x^2 - 148//543*x*y + 532//543*x + y^2 + 182//181*y - 49//543, -148//543*x + 2*y + 182//181, QQMPolyRingElem[440//543*x^2 - 580//543*x*y - 44//543*x + 136//181*y + 112//543, 80//181*x^2 + 320//543*x*y + 122//181*x + 81//181*y + 49//543, -460//543*x^2 - 10//181*x*y - 418//543*x + 32//181*y + 56//181])