Implement univariate polynomial rings over absolute number fields

[lftabera]

After some discussion with Sebastian Spancratz it can be interesting to add a specific implementation for univariate polynomials over number fields. This can improve a lot performance, at least for multiplication, addition and gcd.

One approach is to implement Nuberfield(f)[x] more likely QQ[x][y].quotient(f(y))

Note, with patch #10255

sage: K=QQ[x]['y']
sage: y=K.gen()
sage: L=K.quotient(y^16+y^5+y^4+y^3+y^2+y+1)
sage: f=L(K.random_element(16,1500))
sage: g=L(K.random_element(16,1500))
sage: P=NumberField(x^16+x^5+x^4+x^3+x^2+x+1,'a')[x]
sage: f1 = P.random_element(1500)
sage: g1 = P.random_element(1500)
sage: def nfpol_to_pari(f):
    return pari([c._pari_('a') for c in f.list()]).Polrev()
....: 
sage: fpari = nfpol_to_pari(f1)
sage: gpari = nfpol_to_pari(g1)
sage: %time _ = f*g
CPU times: user 1.92 s, sys: 0.00 s, total: 1.92 s
Wall time: 1.94 s
sage: %time _ = f1*g1
CPU times: user 20.29 s, sys: 0.04 s, total: 20.32 s
Wall time: 20.34 s
sage: %time _ = fpari*gpari
CPU times: user 66.50 s, sys: 0.02 s, total: 66.52 s
Wall time: 66.58 s
sage: %time _=f+g
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.01 s
sage: %time _=f1+g1
CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
Wall time: 0.02 s
sage: %time _=fpari+gpari
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s

Related tickets: #8558, #10255

Things to do (to be completed):

• Faster multiplication
• Fast modular gcd add a fast gcd algorithm for univariate polynomials over absolute number fields #8558
• Avoid PARI nfinit to compute factorization
• Try Newton-based quo_rem

CC: @wbhart @miguelmarco

Component: basic arithmetic

Keywords: polynomials

Issue created by migration from https://trac.sagemath.org/ticket/10591

[miguelmarco]

comment:8

Flint should have suppport for this through Antic, right?