def cyclotomic_to_gamma(cyclo_up, cyclo_down):
"""
Convert a quotient of products of cyclotomic polynomials
to a quotient of products of polynomials `x^n - 1`.
INPUT:
- ``cyclo_up`` -- list of indices of cyclotomic polynomials in the numerator
- ``cyclo_down`` -- list of indices of cyclotomic polynomials in the denominator
OUTPUT:
a dictionary mapping an integer `n` to the power of `x^n - 1` that
appears in the given product
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import cyclotomic_to_gamma
sage: cyclotomic_to_gamma([6], [1])
{2: -1, 3: -1, 6: 1}
"""
dico = defaultdict(int)
for d in cyclo_up:
dico[d] += 1
for d in cyclo_down:
dico[d] -= 1
resu = defaultdict(int)
for n in dico:
for d in divisors(n):
resu[d] += moebius(n / d) * dico[n]
return {d: resu[d] for d in resu if resu[d]}
def cyclotomic_to_gamma(cyclo_up, cyclo_down):
"""
Convert a quotient of products of cyclotomic polynomials
to a quotient of products of polynomials `x^n - 1`.
INPUT:
- ``cyclo_up`` -- list of indices of cyclotomic polynomials in the numerator
- ``cyclo_down`` -- list of indices of cyclotomic polynomials in the denominator
OUTPUT:
a dictionary mapping an integer `n` to the power of `x^n - 1` that
appears in the given product
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import cyclotomic_to_gamma
sage: cyclotomic_to_gamma([6], [1])
{2: -1, 3: -1, 6: 1}
"""
dico = defaultdict(int)
for d in cyclo_up:
dico[d] += 1
for d in cyclo_down:
dico[d] -= 1
resu = defaultdict(int)
for n in dico:
for d in divisors(n):
resu[d] += moebius(n / d) * dico[n]
return {d: resu[d] for d in resu if resu[d]}
def capital_M(n):
"""
Auxiliary function, used to describe the canonical scheme.
INPUT:
- ``n`` -- an integer
OUTPUT:
a rational
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import capital_M
sage: [capital_M(i) for i in range(1,8)]
[1, 4, 27, 64, 3125, 432, 823543]
"""
n = ZZ(n)
return QQ.prod(d ** (d * moebius(n / d)) for d in divisors(n))
def capital_M(n):
"""
Auxiliary function, used to describe the canonical scheme.
INPUT:
- ``n`` -- an integer
OUTPUT:
a rational
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import capital_M
sage: [capital_M(i) for i in range(1,8)]
[1, 4, 27, 64, 3125, 432, 823543]
"""
n = ZZ(n)
return QQ.prod(d ** (d * moebius(n / d)) for d in divisors(n))
