Enumeration of primitive totally real fields

This module contains functions for enumerating all primitive totally real number fields of given degree and small discriminant. Here a number field is called primitive if it contains no proper subfields except \(\QQ\).

See also sage.rings.number_field.totallyreal_rel, which handles the non-primitive case using relative extensions.

ALGORITHM:

We use Hunter’s algorithm ([Coh2000], Section 9.3) with modifications due to Takeuchi [Tak1999] and the author [Voi2008].

We enumerate polynomials \(f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0\). Hunter’s theorem gives bounds on \(a_{n-1}\) and \(a_{n-2}\); then given \(a_{n-1}\) and \(a_{n-2}\), one can recursively compute bounds on \(a_{n-3}, \dots, a_0\), using the fact that the polynomial is totally real by looking at the zeros of successive derivatives and applying Rolle’s theorem. See [Tak1999] for more details.

EXAMPLES:

In this first simple example, we compute the totally real quadratic fields of discriminant \(\le 50\).

sage: enumerate_totallyreal_fields_prim(2,50)
[[5, x^2 - x - 1],
 [8, x^2 - 2],
 [12, x^2 - 3],
 [13, x^2 - x - 3],
 [17, x^2 - x - 4],
 [21, x^2 - x - 5],
 [24, x^2 - 6],
 [28, x^2 - 7],
 [29, x^2 - x - 7],
 [33, x^2 - x - 8],
 [37, x^2 - x - 9],
 [40, x^2 - 10],
 [41, x^2 - x - 10],
 [44, x^2 - 11]]
sage: [d for d in range(5,50)
....:    if (is_squarefree(d) and d%4 == 1) or (d%4 == 0 and is_squarefree(d/4))]
[5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 33, 37, 40, 41, 44]

>>> from sage.all import *
>>> enumerate_totallyreal_fields_prim(Integer(2),Integer(50))
[[5, x^2 - x - 1],
 [8, x^2 - 2],
 [12, x^2 - 3],
 [13, x^2 - x - 3],
 [17, x^2 - x - 4],
 [21, x^2 - x - 5],
 [24, x^2 - 6],
 [28, x^2 - 7],
 [29, x^2 - x - 7],
 [33, x^2 - x - 8],
 [37, x^2 - x - 9],
 [40, x^2 - 10],
 [41, x^2 - x - 10],
 [44, x^2 - 11]]
>>> [d for d in range(Integer(5),Integer(50))
...    if (is_squarefree(d) and d%Integer(4) == Integer(1)) or (d%Integer(4) == Integer(0) and is_squarefree(d/Integer(4)))]
[5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 33, 37, 40, 41, 44]

enumerate_totallyreal_fields_prim(2,50)
[d for d in range(5,50)
   if (is_squarefree(d) and d%4 == 1) or (d%4 == 0 and is_squarefree(d/4))]

Next, we compute all totally real quintic fields of discriminant \(\le 10^5\):

sage: ls = enumerate_totallyreal_fields_prim(5,10^5) ; ls
[[14641, x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1],
 [24217, x^5 - 5*x^3 - x^2 + 3*x + 1],
 [36497, x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1],
 [38569, x^5 - 5*x^3 + 4*x - 1],
 [65657, x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1],
 [70601, x^5 - x^4 - 5*x^3 + 2*x^2 + 3*x - 1],
 [81509, x^5 - x^4 - 5*x^3 + 3*x^2 + 5*x - 2],
 [81589, x^5 - 6*x^3 + 8*x - 1],
 [89417, x^5 - 6*x^3 - x^2 + 8*x + 3]]
 sage: len(ls)
 9

>>> from sage.all import *
>>> ls = enumerate_totallyreal_fields_prim(Integer(5),Integer(10)**Integer(5)) ; ls
[[14641, x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1],
 [24217, x^5 - 5*x^3 - x^2 + 3*x + 1],
 [36497, x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1],
 [38569, x^5 - 5*x^3 + 4*x - 1],
 [65657, x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1],
 [70601, x^5 - x^4 - 5*x^3 + 2*x^2 + 3*x - 1],
 [81509, x^5 - x^4 - 5*x^3 + 3*x^2 + 5*x - 2],
 [81589, x^5 - 6*x^3 + 8*x - 1],
 [89417, x^5 - 6*x^3 - x^2 + 8*x + 3]]
>>> len(ls)
 9

ls = enumerate_totallyreal_fields_prim(5,10^5) ; ls
len(ls)

We see that there are 9 such fields (up to isomorphism!).

See also [Mar1980].

AUTHORS:

  • John Voight (2007-09-01): initial version; various optimization tweaks

  • John Voight (2007-10-09): added DSage module; added pari functions to avoid recomputations; separated DSage component

  • Craig Citro and John Voight (2007-11-04): additional doctests and type checking

  • Craig Citro and John Voight (2008-02-10): final modifications for submission

sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim(n, B, a=[], verbose=0, return_seqs=False, phc=False, keep_fields=False, t_2=False, just_print=False, return_pari_objects=True)[source]

Enumerate primitive totally real fields of degree \(n>1\) with discriminant \(d \leq B\); optionally one can specify the first few coefficients, where the sequence \(a\) corresponds to

a[d]*x^n + ... + a[0]*x^(n-d)

where length(a) = d+1, so in particular always a[d] = 1.

Note

This is guaranteed to give all primitive such fields, and seems in practice to give many imprimitive ones.

INPUT:

  • n – integer; the degree

  • B – integer; the discriminant bound

  • a – list (default: []); the coefficient list to begin with

  • verbose – (integer or string, default: 0) if verbose == 1 (or 2), then print to the screen (really) verbosely; if verbose is a string, then print verbosely to the file specified by verbose.

  • return_seqs – boolean (default: False); if True, then return the polynomials as sequences (for easier exporting to a file)

  • phc – boolean or integer (default: False)

  • keep_fields – boolean or integer (default: False); if keep_fields is True, then keep fields up to B*log(B); if keep_fields is an integer, then keep fields up to that integer.

  • t_2 – boolean or integer (default: False); if t_2 = T, then keep only polynomials with t_2 norm >= T

  • just_print – boolean (default: False); if just_print is not False, instead of creating a sorted list of totally real number fields, we simply write each totally real field we find to the file whose filename is given by just_print. In this case, we don’t return anything.

  • return_pari_objects – boolean (default: True); if both return_seqs and return_pari_objects are False then it returns the elements as Sage objects. Otherwise it returns PARI objects.

OUTPUT:

the list of fields with entries [d,f], where d is the discriminant and f is a defining polynomial, sorted by discriminant.

AUTHORS:

  • John Voight (2007-09-03)

  • Craig Citro (2008-09-19): moved to Cython for speed improvement

sage.rings.number_field.totallyreal.odlyzko_bound_totallyreal(n)[source]

This function returns the unconditional Odlyzko bound for the root discriminant of a totally real number field of degree \(n\).

Note

The bounds for \(n > 50\) are not necessarily optimal.

INPUT:

  • n – integer; the degree

OUTPUT: a lower bound on the root discriminant (as a real number)

EXAMPLES:

sage: from sage.rings.number_field.totallyreal import odlyzko_bound_totallyreal
sage: [odlyzko_bound_totallyreal(n) for n in range(1, 5)]
[1.0, 2.223, 3.61, 5.067]

>>> from sage.all import *
>>> from sage.rings.number_field.totallyreal import odlyzko_bound_totallyreal
>>> [odlyzko_bound_totallyreal(n) for n in range(Integer(1), Integer(5))]
[1.0, 2.223, 3.61, 5.067]

from sage.rings.number_field.totallyreal import odlyzko_bound_totallyreal
[odlyzko_bound_totallyreal(n) for n in range(1, 5)]

AUTHORS:

  • John Voight (2007-09-03)

Note

The values are calculated by Martinet [Mar1980].

sage.rings.number_field.totallyreal.weed_fields(S, lenS=0)[source]

Function used internally by the enumerate_totallyreal_fields_prim() routine. (Weeds the fields listed by [discriminant, polynomial] for isomorphism classes.) Returns the size of the resulting list.

EXAMPLES:

sage: ls = [[5,pari('x^2-3*x+1')],[5,pari('x^2-5')]]
sage: sage.rings.number_field.totallyreal.weed_fields(ls)
1
sage: ls
[[5, x^2 - 3*x + 1]]

>>> from sage.all import *
>>> ls = [[Integer(5),pari('x^2-3*x+1')],[Integer(5),pari('x^2-5')]]
>>> sage.rings.number_field.totallyreal.weed_fields(ls)
1
>>> ls
[[5, x^2 - 3*x + 1]]

ls = [[5,pari('x^2-3*x+1')],[5,pari('x^2-5')]]
sage.rings.number_field.totallyreal.weed_fields(ls)
ls