Enumeration of Totally Real Fields¶
AUTHORS:
Craig Citro and John Voight (2007-11-04): Type checking and other polishing.
John Voight (2007-10-09): Improvements: Smyth bound, Lagrange multipliers for b.
John Voight (2007-09-19): Various optimization tweaks.
John Voight (2007-09-01): Initial version.
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sage.rings.number_field.totallyreal_data.easy_is_irreducible_py(f)¶ Used solely for testing easy_is_irreducible.
EXAMPLES:
sage: sage.rings.number_field.totallyreal_data.easy_is_irreducible_py(pari('x^2+1')) 1 sage: sage.rings.number_field.totallyreal_data.easy_is_irreducible_py(pari('x^2-1')) 0
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sage.rings.number_field.totallyreal_data.hermite_constant(n)¶ This function returns the nth Hermite constant
The nth Hermite constant (typically denoted \(\gamma_n\)), is defined to be
\[\max_L \min_{0 \neq x \in L} ||x||^2\]where \(L\) runs over all lattices of dimension \(n\) and determinant \(1\).
For \(n \leq 8\) it returns the exact value of \(\gamma_n\), and for \(n > 9\) it returns an upper bound on \(\gamma_n\).
INPUT:
n – integer
OUTPUT:
(an upper bound for) the Hermite constant gamma_n
EXAMPLES:
sage: hermite_constant(1) # trivial one-dimensional lattice 1.0 sage: hermite_constant(2) # Eisenstein lattice 1.1547005383792515 sage: 2/sqrt(3.) 1.15470053837925 sage: hermite_constant(8) # E_8 2.0
AUTHORS:
John Voight (2007-09-03)
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sage.rings.number_field.totallyreal_data.int_has_small_square_divisor(d)¶ Returns the largest a such that a^2 divides d and a has prime divisors < 200.
EXAMPLES:
sage: from sage.rings.number_field.totallyreal_data import int_has_small_square_divisor sage: int_has_small_square_divisor(500) 100 sage: is_prime(691) True sage: int_has_small_square_divisor(691) 1 sage: int_has_small_square_divisor(691^2) 1
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sage.rings.number_field.totallyreal_data.lagrange_degree_3(n, an1, an2, an3)¶ Private function. Solves the equations which arise in the Lagrange multiplier for degree 3: for each 1 <= r <= n-2, we solve
r*x^i + (n-1-r)*y^i + z^i = s_i (i = 1,2,3)
where the s_i are the power sums determined by the coefficients a. We output the largest value of z which occurs. We use a precomputed elimination ideal.
EXAMPLES:
sage: ls = sage.rings.number_field.totallyreal_data.lagrange_degree_3(3,0,1,2) sage: [RealField(10)(x) for x in ls] [-1.0, -1.0] sage: sage.rings.number_field.totallyreal_data.lagrange_degree_3(3,6,1,2) # random [-5.8878, -5.8878]
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class
sage.rings.number_field.totallyreal_data.tr_data¶ Bases:
objectThis class encodes the data used in the enumeration of totally real fields.
We do not give a complete description here. For more information, see the attached functions; all of these are used internally by the functions in totallyreal.py, so see that file for examples and further documentation.
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increment(verbose=False, haltk=0, phc=False)¶ This function ‘increments’ the totally real data to the next value which satisfies the bounds essentially given by Rolle’s theorem, and returns the next polynomial as a sequence of integers.
The default or usual case just increments the constant coefficient; then inductively, if this is outside of the bounds we increment the next higher coefficient, and so on.
If there are no more coefficients to be had, returns the zero polynomial.
INPUT:
verbose – boolean to print verbosely computational details
haltk – integer, the level at which to halt the inductive coefficient bounds
phc – boolean, if PHCPACK is available, use it when k == n-5 to compute an improved Lagrange multiplier bound
OUTPUT:
The next polynomial, as a sequence of integers
EXAMPLES:
sage: T = sage.rings.number_field.totallyreal_data.tr_data(2,100) sage: T.increment() [-24, -1, 1] sage: for i in range(19): _ = T.increment() sage: T.increment() [-3, -1, 1] sage: T.increment() [-25, 0, 1]
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printa()¶ Print relevant data for self.
EXAMPLES:
sage: T = sage.rings.number_field.totallyreal_data.tr_data(3,2^10) sage: T.printa() k = 1 a = [0, 0, -1, 1] amax = [0, 0, 0, 1] beta = [...] gnk = [...]
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