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]
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
We see that there are 9 such fields (up to isomorphism!).
See also [Mar1980].
- 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)#
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 alwaysa[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 degreeB
– (integer) the discriminant bounda
– (list, default:[]
) the coefficient list to begin withverbose
– (integer or string, default: 0) ifverbose == 1
(or2
), then print to the screen (really) verbosely; if verbose is a string, then print verbosely to the file specified by verbose.return_seqs
– (boolean, defaultFalse
) IfTrue
, 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
) Ifkeep_fields
isTrue
, then keep fields up toB*log(B)
; ifkeep_fields
is an integer, then keep fields up to that integer.t_2
– (boolean or integer, default: False) Ift_2 = T
, then keep only polynomials with t_2 norm >= T.just_print
– (boolean, default:False
): ifjust_print
is notFalse
, 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 byjust_print
. In this case, we don’t return anything.return_pari_objects
– (boolean, default:True
) if bothreturn_seqs
andreturn_pari_objects
areFalse
then it returns the elements as Sage objects; otherwise it returns PARI objects.
OUTPUT:
the list of fields with entries
[d,f]
, whered
is the discriminant andf
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)#
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]
AUTHORS:
John Voight (2007-09-03)
Note
The values are calculated by Martinet [Mar1980].
- sage.rings.number_field.totallyreal.weed_fields(S, lenS=0)#
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]]