msolve

julia> using GroebnerBasis, Singular

julia> R, (a, b, c) = PolynomialRing(QQ, ["a", "b", "c"]);

julia> I(n) = Ideal(R, [(a+b)^n+c-3, (b+c)^n+a-2, (2*c+a)^n+b-1]);

julia> for n = 5:15 print(n,":"); @time msolve(I(n), get_param = true) end;
5:  0.122010 seconds (11.82 k allocations: 1.950 MiB, 0.02% compilation time)
6:  0.348948 seconds (19.29 k allocations: 4.985 MiB, 0.01% compilation time)
7:  0.878888 seconds (27.93 k allocations: 11.589 MiB, 0.01% compilation time)
8:  2.193366 seconds (40.84 k allocations: 24.917 MiB, 0.00% compilation time)
9:  4.885550 seconds (57.68 k allocations: 49.346 MiB, 0.00% compilation time)
10: 11.550713 seconds (82.87 k allocations: 91.559 MiB, 0.06% gc time, 0.00% compilation time)
11: 23.599137 seconds (105.80 k allocations: 160.780 MiB, 0.00% compilation time)
12: 52.038471 seconds (139.22 k allocations: 269.434 MiB, 0.02% gc time, 0.00% compilation time)
13: 99.050393 seconds (253.12 k allocations: 453.029 MiB, 0.02% gc time, 0.10% compilation time)
14:202.253239 seconds (331.56 k allocations: 691.498 MiB, 0.10% gc time, 0.07% compilation time)
15:362.278998 seconds (358.05 k allocations: 1.065 GiB, 0.01% gc time, 0.03% compilation time)

julia> using GroebnerBasis, Singular, Nemo;

julia> # problem
       svs = ["a", "b", "c"]; sys((a, b, c)) = [(a+b)^5+c-3,(b+c)^5+a-2,(2*c+a)^5+b-1];

julia> R, vs = Singular.PolynomialRing(Singular.QQ, svs);

julia>  # run msolve
       @time prm, nmc = msolve(Ideal(R, sys(vs)), get_param = true);
  3.348726 seconds (3.36 M allocations: 204.432 MiB, 1.05% gc time, 96.05% compilation time)

julia> p1, _, p3, p4, p5, p6 = prm;

julia> # vars. order
       pmt = map(s->findfirst(e->e==s, p1), svs);

julia> # for factors
       fcs = factor(p3); println("#irr_fct_elm: ", length(fcs))
#irr_fct_elm: 1

julia> for (fct, _) in fcs
       print("deg: ", Singular.degree(fct), "\nrur time:");
        # rur using nf_elem
       p = Nemo.NumberField(fct, "p")[2];
       @time rur = vcat([Nemo.evaluate((-1)*e1, p)//Nemo.evaluate(e2*p4, p) for (e1, e2) in zip(p5, p6)], [p]);
        # Symbolic test
       @time println(sys(rur[pmt]));
        # Numeric test for raw
       map(e->println(sys(map(Float64, e[pmt]))), nmc);
        # Numeric test for solve_rational_parametrization
       @time srp = solve_rational_parametrization(prm[3:end], precision = 100);
       map(e->println(sys(map(Float64, e[pmt]))), srp);
        end;
deg: 125
rur time:  1.995804 seconds (1.41 M allocations: 171.327 MiB, 1.16% gc time, 23.01% compilation time)
nf_elem[0, 0, 0]
  0.745820 seconds (642.35 k allocations: 196.643 MiB, 1.31% gc time, 55.69% compilation time)
[-2.220446049250313e-15, -8.881784197001252e-16, 0.0]
den: fmpq_poly
  2.880423 seconds (3.58 M allocations: 205.021 MiB, 1.46% gc time, 0.58% compilation time)
[-2.220446049250313e-15, -8.881784197001252e-16, 0.0]

? msolve2("kat10-qq");
--2021-04-20 22:46:37--  https://gitlab.lip6.fr/eder/msolve-examples/-/raw/master/zero-dimensional/kat10-qq.ms
Resolving gitlab.lip6.fr (gitlab.lip6.fr)... 132.227.201.130
Connecting to gitlab.lip6.fr (gitlab.lip6.fr)|132.227.201.130|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 665 [text/plain]
Saving to: 'kat10-qq.ms'

kat10-qq.ms                  100%[===========================================>]     665  --.-KB/s    in 0s      

2021-04-20 22:46:39 (36.0 MB/s) - 'kat10-qq.ms' saved [665/665]

26.73user 0.01system 0:26.77elapsed 99%CPU (0avgtext+0avgdata 21152maxresident)k
0inputs+8144outputs (0major+15298minor)pagefaults 0swaps
120[[0,0,0,0,0,0,0,0,0,0]]
time = 271 ms.

Tst.jl

using GroebnerBasis, Singular, Nemo; 

# for global R, I
function tst1()
    vs = gens(R);
    sys(lst) = (
        map(e->eval(Meta.parse("$(e[1]) = $(e[2])")), zip(vs, lst)); 
        map(e->eval(Meta.parse("$e")), gens(I))
        );
    # run msolve
    print("msolve with param: ");
    @time prm, nmc = msolve(I, get_param = true);
    # vars. order
    p1 = prm[1]; println(p1);
    pmt = map(s->findfirst(e->e==string(s), p1), vs);
    # tests
    tst0(sys, prm, nmc, pmt);
end;

# for global svs, sys
function tst2()
    R, vs = Singular.PolynomialRing(Singular.QQ, svs);
    # run msolve
    print("msolve with param: ");
    @time prm, nmc = msolve(Ideal(R, sys(vs)), get_param = true);
    # vars. order
    p1 = prm[1]; println(p1);
    pmt = map(s->findfirst(e->e==s, p1), svs);
    # tests
    tst0(sys, prm, nmc, pmt);
end;

# rur and tests
function tst0(sys, prm, nmc, pmt)
    p3, p4, p5, p6 = prm[3:end];
    # for factors
    fcs = factor(p3); println("#irr_fct_elm: ", length(fcs));
    for (fct, _) in fcs
        print("deg: ", degree(fct), "\nrur:");
        # rur
        global p = Nemo.NumberField(fct, "p")[2];
        @time global rur = vcat(
            [Nemo.evaluate((-1)*p5[k], p)//Nemo.evaluate(e*p4, p) for (k, e) in enumerate(p6)], 
            [p]
            );
        # Symbolic test
        println("sym:");
        @time println(sys(rur[pmt]));
        # Numeric test for raw
        println("num_raw:");
        map(e->println(sys(map(Float64, e[pmt]))), nmc);
        # Numeric test for solve_rational_parametrization
        println("num_srp:");
        @time srp = solve_rational_parametrization(prm[3:end], precision = 100);
        map(e->println(sys(map(Float64, e[pmt]))), srp);
    end;
end;

#=
# prob.1
R, I = GroebnerBasis.katsura_4(0);
tst1();
# prob.2
svs = ["a", "b", "c"];
sys((a, b, c)) = [-1+c+b+a^3, -1+b+c*a+2*a^3, 1+c*b+c^2*a];
tst2();
# prob.3
R, (a, b, c) = Singular.PolynomialRing(Singular.QQ, ["a", "b", "c"]);
I = Ideal(R, [a^16+b+c, a+b^3+c-2, a*b*c-1]);
for n = 1:50 tst1() end;
# prob.4
svs = ["a", "b", "c"];
for n in [16, 25, 27, 29, 31, 38, 40, 42, 44, 46, 51, 55, 61, 66, 68, 70, 74, 76, 91];
    println(n," >>>");
    global sys((a, b, c)) = [a^n+b+c, a+b^3+c-2, a*b*c-1];
    tst2();
end;
=#

https://arxiv.org/pdf/2104.03572.pdf
https://msolve.lip6.fr/
https://icerm.brown.edu/video_archive/?play=2455