Bug in primary decomposition & does not terminate on a principal ideal

[simonbrandhorst]

Describe the bug Let K be a number field (e.g. a cyclotomic field)

1. Primary decomposition does not terminate after 12 hours for a principal ideal (f) < K[x,y,t] (but factor(f) terminates immediately)

2. Primary decomposition throws an error for K a number field and J an ideal over K(t)[x,y]

To Reproduce

using Oscar
F, a = cyclotomic_field(7,:a)
P, (x,y,t) = polynomial_ring(F, [:x,:y,:t])
f = -x^3 + (127961567541885//2*a^5 + 230578777287775//2*a^4 + 230578777287775//2*a^3 + 127961567541885//2*a^2 - 455586040332045//16)*x + y^2 + (3242437110294043228*a^5 + 5842669785012830924*a^4 + 5842669785012830924*a^3 + 3242437110294043228*a^2 - 1443020268154487519)*t^7 + 9351999235365594180359//8*a^5 + 16851720327423257592547//8*a^4 + 16851720327423257592547//8*a^3 + 9351999235365594180359//8*a^2 - 16648124833698322506703//32
I = ideal(f)
factor(f) # works in seconds and confirms that f is irreducible
primary_decomposition(I) # keeps running for hours

The second case:

using Oscar
# Julia code
F, a = cyclotomic_field(7,:a)
P,x = polynomial_ring(F,:x)
f = x^3 - x^2 - 2*x + 1
w = roots(f)[1]
#F,w = number_field(f,:w)
Ft, t = polynomial_ring(F,:t)
FFt = fraction_field(Ft)
E1 = elliptic_curve(FFt,[0,0,0, (-230578777287775//2*w^2+127961567541885//2*w+4144846476936445//16), ((-5842669785012830924*w^2+3242437110294043228*w+13128359838180149367)*t^7-16851720327423257592547//8*w^2+9351999235365594180359//8*w+151461887453084383247079//32)])

then

julia> J = ideal(equation(E1))
Ideal generated by
  -x^3 + (127961567541885//2*a^5 + 230578777287775//2*a^4 + 230578777287775//2*a^3 + 127
  961567541885//2*a^2 - 455586040332045//16)*x + y^2 + (3242437110294043228*a^5 + 584266
  9785012830924*a^4 + 5842669785012830924*a^3 + 3242437110294043228*a^2 - 14430202681544
  87519)*t^7 + 9351999235365594180359//8*a^5 + 16851720327423257592547//8*a^4 + 16851720
  327423257592547//8*a^3 + 9351999235365594180359//8*a^2 - 16648124833698322506703//32

julia> primary_decomposition(J)
ERROR: Unexpected case in prepareQuotientring. Please inform the authors
leaving primdec.lib::prepareQuotientring (0)
leaving primdec.lib::decomp_i (3361)
leaving primdec.lib::primdecGTZ_i (5677)

Stacktrace:
  [1] call_singular_library_procedure
    @ ~/.julia/packages/CxxWrap/5IZvn/src/CxxWrap.jl:624 [inlined]
  [2] low_level_caller_rng(lib::String, name::String, ring::Singular.PolyRing{…}, args::Singular.sideal{…})
    @ Singular ~/.julia/packages/Singular/YQAWd/src/caller.jl:379
  [3] primdecGTZ(ring::Singular.PolyRing{Singular.n_FieldElem{…}}, args::Singular.sideal{Singular.spoly{…}})

Expected behavior The function should work and terminate as described in the docu :-).

System (please complete the following information):

OSCAR version 1.2.0-DEV - #sb/EnriquesAut, d92758d3a2 -- 2024-07-03 09:53:15 +0000
  combining:
    AbstractAlgebra.jl   v0.41.9
    GAP.jl               v0.10.4
    Hecke.jl             v0.32.3 - #master, 0cf400dfaa -- 2024-06-13 08:05:41 +0200
    Nemo.jl              v0.45.5
    Polymake.jl          v0.11.18
    Singular.jl          v0.23.2
  building on:
    FLINT_jll               v300.100.300+0
    GAP_jll                 v400.1200.200+9
    Singular_jll            v404.0.100+0
    libpolymake_julia_jll   v0.12.0+0
    libsingular_julia_jll   v0.45.1+0
    polymake_jll            v400.1200.1+0
See `]st -m` for a full list of dependencies.

Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 20 × 13th Gen Intel(R) Core(TM) i7-1370P
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, goldmont)
Threads: 15 default, 0 interactive, 1 GC (on 20 virtual cores)
Environment:
  JULIA_NUM_GC_THREADS = 1
  JULIA_NUM_THREADS = 15
Official https://julialang.org/ release

Additional context Add any other context about the problem here.