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Optimize local fields a bit (thofma#1567)
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* Don't multiply the precision with the ramification index all the time

* Store `absolute_ramification_index`

* Avoid some copies

* Add valuation onto ZZ and speed up `mul!` a bit

* `numerator(divexact(v, degree(K))*ramification_index(K)) ==
numerator(v)`

* Use the correct precision in `a - 1`
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@joschmitt
joschmitt authored Aug 6, 2024
1 parent da36757 commit a503506
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Showing 4 changed files with 109 additions and 99 deletions.
2 changes: 1 addition & 1 deletion src/LocalField/Completions.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -461,7 +461,7 @@ function setprecision!(f::CompletionMap{LocalField{PadicFieldElem, EisensteinLoc
coeffs_eisenstein[e+1] = one(Qp)
pol_gen = Qpx(coeffs_eisenstein)
Kp.def_poly_cache[new_prec] = pol_gen
Kp.precision = new_prec
setprecision!(Kp, new_prec * ramification_index(Kp))
#Should I update the traces too?
img_prim_elem = Vector{PadicFieldElem}(undef, e)
for i = 1:e
Expand Down
176 changes: 85 additions & 91 deletions src/LocalField/Elem.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -35,23 +35,17 @@ end
#
################################################################################

function _generator_valuation(K::LocalField{S, T}) where {S <: Union{PadicFieldElem, QadicFieldElem}, T <: LocalFieldParameter}
f = defining_polynomial(K)
# @show K, f
return QQFieldElem(valuation(coeff(f, 0)), degree(f))
end

function _generator_valuation(K::LocalField)
f = defining_polynomial(K)
return divexact(valuation(coeff(f, 0)), degree(K))
end

function compute_precision(K::LocalField, a::Generic.Poly)
v = numerator(_generator_valuation(K)*ramification_index(K))
prec = precision(coeff(a, 0))*ramification_index(K)
degree(a) == 0 && return prec
f = defining_polynomial(K)
v = Int(numerator(valuation(coeff(f, 0))//degree(K) * ramification_index(K)))
# TODO: Presumably, v is supposed to be the valuation of gen(K). Is this correct?
vi = 0
for i = 1:degree(a)
vi += v
c = coeff(a, i)
prec = min(prec, precision(c)*ramification_index(K)+Int(numerator(ceil(i*v))))
prec = min(prec, precision(c)*ramification_index(K) + vi)
end
return prec
end
Expand Down Expand Up @@ -101,8 +95,9 @@ parent_type(::Type{LocalFieldElem{S, T}}) where {S <: FieldElem, T <: LocalField
#
################################################################################

coeff(a::LocalFieldElem, i::Int) = coeff(a.data, i)
setcoeff!(a::LocalFieldElem, i::Int) = setcoeff!(a.data, i)
data(a::LocalFieldElem) = a.data
coeff(a::LocalFieldElem, i::Int) = coeff(data(a), i)
setcoeff!(a::LocalFieldElem, i::Int) = setcoeff!(data(a), i)

################################################################################
#
Expand Down Expand Up @@ -227,8 +222,12 @@ function iszero(a::LocalFieldElem{S, EisensteinLocalField}) where S
return true
end

isone(a::LocalFieldElem) = isone(a.data)
isone(a::LocalFieldElem{S, EisensteinLocalField}) where S = iszero(a-1)
isone(a::LocalFieldElem) = isone(data(a))
function isone(a::LocalFieldElem{S, EisensteinLocalField}) where S
a1 = a - one(parent(a), precision = precision(a))
return iszero(a1)
end

is_unit(a::LocalFieldElem) = !iszero(a)

function O(K::LocalField, prec::T) where T <: IntegerUnion
Expand All @@ -241,19 +240,19 @@ end

function zero(K::LocalField; precision=precision(K))
a = zero(parent(defining_polynomial(K)))
return setprecision(K(a), precision)
return setprecision!(K(a), precision)
end

(K::LocalField)(; precision=precision(K)) = zero(K, precision = precision)

function one(K::LocalField; precision=precision(K))
a = one(parent(defining_polynomial(K)))
return setprecision(K(a), precision)
return setprecision!(K(a), precision)
end

function zero!(a::LocalFieldElem; precision=precision(parent(a)))
zero!(a.data)
a.data = setprecision(a.data, precision)
a.data = setprecision!(a.data, precision)
return a
end

Expand Down Expand Up @@ -343,36 +342,46 @@ function valuation(a::LocalFieldElem{S, T}) where {S <: FieldElem, T <: LocalFie
return valuation(norm(a))//degree(parent(a))
end

function valuation(a::LocalFieldElem{S, UnramifiedLocalField}) where S <: FieldElem
return QQ(_valuation_integral(a), absolute_ramification_index(base_field(parent(a))))
end

function valuation(a::LocalFieldElem{S, EisensteinLocalField}) where S <: FieldElem
return QQ(_valuation_integral(a), absolute_ramification_index(parent(a)))
end

_valuation_integral(a::Union{PadicFieldElem, QadicFieldElem}) = valuation(a)

function _valuation_integral(a::LocalFieldElem)
return numerator(absolute_ramification_index(parent(a))*valuation(a))
end

function _valuation_integral(a::LocalFieldElem{S, UnramifiedLocalField}) where S <: FieldElem
if iszero(a)
error("Input element is zero")
end
K = parent(a)
e = absolute_ramification_index(K)
i = 0
c = coeff(a, i)
while iszero(c)
i += 1
i > degree(parent(a)) && error("interesting element")
c = coeff(a, i)
end
vc = valuation(c)
v = vc + QQFieldElem(i, e)
for j = i+1:degree(K)
v = _valuation_integral(c)
for j in i + 1:degree(K)
c = coeff(a, j)
if iszero(c)
continue
end
vc = valuation(c)
vnew = vc + QQFieldElem(j, e)
if vnew < v
v = vnew
vc = _valuation_integral(c)
if vc < v
v = vc
end
end
return v
end

function valuation(a::LocalFieldElem{S, UnramifiedLocalField}) where S <: FieldElem
function _valuation_integral(a::LocalFieldElem{S, EisensteinLocalField}) where S <: FieldElem
if iszero(a)
error("Input element is zero")
end
Expand All @@ -382,17 +391,18 @@ function valuation(a::LocalFieldElem{S, UnramifiedLocalField}) where S <: FieldE
c = coeff(a, i)
while iszero(c)
i += 1
i > degree(parent(a)) && error("interesting element")
c = coeff(a, i)
end
v = valuation(c)
for j = i+1:degree(K)
v = e*_valuation_integral(c) + i
for j = i + 1:degree(K)
c = coeff(a, j)
if iszero(c)
continue
end
vc = valuation(c)
if vc < v
v = vc
vnew = e*_valuation_integral(c) + j
if vnew < v
v = vnew
end
end
return v
Expand Down Expand Up @@ -563,7 +573,7 @@ end
function Base.:+(a::LocalFieldElem{S, T}, b::LocalFieldElem{S, T}) where {S <: FieldElem, T <: LocalFieldParameter}
check_parent(a, b)
K = parent(a)
c = setprecision(base_ring(a.data), ceil(Int, precision(K)/ramification_index(K))) do
c = setprecision(base_ring(a.data), _precision_base(K)) do
a.data + b.data
end
return LocalFieldElem{S, T}(parent(a), c, min(precision(a), precision(b)))
Expand All @@ -572,7 +582,7 @@ end
function Base.:-(a::LocalFieldElem{S, T}, b::LocalFieldElem{S, T}) where {S <: FieldElem, T <: LocalFieldParameter}
check_parent(a, b)
K = parent(a)
c = setprecision(base_ring(a.data), ceil(Int, precision(K)/ramification_index(K))) do
c = setprecision(base_ring(a.data), _precision_base(K)) do
a.data - b.data
end
return LocalFieldElem{S, T}(parent(a), c, min(precision(a), precision(b)))
Expand Down Expand Up @@ -624,25 +634,14 @@ end

function mul!(c::LocalFieldElem{S, T}, a::LocalFieldElem{S, T}, b::LocalFieldElem{S, T}) where {S <: FieldElem, T <: LocalFieldParameter}
check_parent(a, b)
K = c.parent = a.parent
e = ramification_index(parent(a))
c.data = mul!(c.data, a.data, b.data)
c.data = mod(c.data, defining_polynomial(parent(a), max(precision(c.data), ceil(Int, precision(K)/e))))
# c.precision = compute_precision(a.parent, c.data)
e = absolute_ramification_index(parent(a))
if iszero(a)
va = 0
else
va = Int(e*valuation(a))
end
if iszero(b)
vb = 0
else
vb = Int(e*valuation(b))
end
pr = min(precision(a) - va, precision(b) - vb) + va+vb
c.precision = min(compute_precision(a.parent, c.data), pr)
# c.precision = compute_precision(a.parent, c.data)
K = parent(a)
e = ramification_index(K)
c.data = mul!(c.data, data(a), data(b))
c.data = mod(c.data, defining_polynomial(K, max(precision(data(c)), _precision_base(K))))
va = (iszero(a) ? 0 : Int(_valuation_integral(a)))
vb = (iszero(b) ? 0 : Int(_valuation_integral(b)))
pr = min(precision(a) + vb, precision(b) + va)
c.precision = min(compute_precision(K, data(c)), pr)
return c
end

Expand Down Expand Up @@ -682,8 +681,7 @@ end

function inv(a::LocalFieldElem)
K = parent(a)
e = absolute_ramification_index(parent(a))
v = Int(e*valuation(a))
v = Int(_valuation_integral(a))
if v == 0
p = invmod(a.data, defining_polynomial(K, precision(a.data)))
return K(p)
Expand All @@ -709,17 +707,15 @@ function Base.:(^)(a::LocalFieldElem, n::Int)
if n < 0 && iszero(a)
error("Element is not invertible")
end
e = absolute_ramification_index(parent(a))
v = valuation(n, prime(parent(a)))
v = valuation(n, prime(K))
prec = precision(data(a)) + v
if v > 0
b = setprecision(base_ring(a.data), precision(a.data)+v) do
setprecision(a.data, precision(a.data)+v)
end
b = setprecision(data(a), prec)
else
b = a.data
b = data(a)
end
b = setprecision(base_ring(b), precision(b)) do
powermod(b, n, defining_polynomial(K, precision(b)))
b = setprecision(base_ring(b), prec) do
powermod(b, n, defining_polynomial(K, prec))
end
return K(b)
end
Expand All @@ -739,7 +735,7 @@ function _underlying_base_field(K::T) where T <: Union{PadicField, QadicField}
end

@doc raw"""
log(a::LocalFieldElem) -> LocalFieldElem
exp(a::LocalFieldElem) -> LocalFieldElem
Computes the $p$-adic exponential of $a$.
"""
Expand All @@ -755,11 +751,9 @@ function exp(a::LocalFieldElem)
a = setprecision(a, N)
oN = precision(parent(a))
setprecision!(parent(a), max(oN, N))
res = one(K)
res = setprecision(res, N)
res = one(K, precision = N)
el = one(K)
res = res
den = setprecision!(one(Qp), N)
den = one(Qp, precision = N)
#precision is suboptimal, its truncated badly, thus losing it
max_i = QQFieldElem(N)//(valuation(a) - QQFieldElem(1, p-1)) + 1
bound = Int(floor(ZZRingElem, max_i))
Expand Down Expand Up @@ -796,27 +790,27 @@ function log(a::LocalFieldElem)
e = absolute_ramification_index(K)
f = absolute_inertia_degree(K)
p = prime(K)
pf1 = p^f - 1
pi = uniformizer(K)
y = a*pi^(-Int(numerator(va*e)))
#Now, y has valuation 0
z = y^(p^f-1)
z = y^pf1
#Now, z is a 1-unit
logy = _log_one_units(z)//(p^f-1)
logy = _log_one_units(z)//pf1
eps = ((pi^e)//p)
#Same trick to make eps is now a 1-unit.
if !isone(eps) && iszero(valuation(eps-1))
logeps = divexact(_log_one_units(eps^(p^f-1)), p^f-1)
logeps = divexact(_log_one_units(eps^pf1), pf1)
else
logeps = _log_one_units(eps)
end
return logy + va*logeps
end


function _log_one_units(a::LocalFieldElem)
K = parent(a)
if isone(a)
return setprecision!(zero(K), precision(a))
return zero(K, precision = precision(a))
end
#TODO: computing log(a^p^l)//p^l might accelerate the
#computation. Find the optimal l.
Expand All @@ -826,30 +820,30 @@ function _log_one_units(a::LocalFieldElem)
p = prime(K)
el = deepcopy(a)
d = ZZRingElem(1)
e = absolute_ramification_index(K)
v = numerator(e*valuation(a-1))
v = _valuation_integral(a - one(K, precision = precision(a)))
N = precision(el)
num = a
den = ZZRingElem(1)
candidate = div(N, v)
#XXX: currently broke!
# the precision is not increasing...
while true
Nv = candidate
while Nv != 1
d *= p
el = el^p
N = precision(el)
if isone(el) || iszero(el - 1)
el1 = el - one(K, precision = N)

if iszero(el1)
num = el
den = d
break
end
attempt = clog(d, 2) + div(N, numerator(e*valuation(el-1)))
if attempt > candidate
break
else
num = el
den = d
end

Nv = div(N, _valuation_integral(el1))
attempt = clog(d, 2) + Nv
attempt > candidate && break

num = el
den = d
end
r = _log_one_units_fast(num)
r = divexact(r, den)
Expand All @@ -864,7 +858,7 @@ function divexact(a::LocalFieldElem, b::Union{Integer, ZZRingElem}; check::Bool=
iszero(a) && return setprecision(a, precision(a) - v*e)
Qp = prime_field(K)
old = precision(Qp)
setprecision!(Qp, e*precision(a)+round(Int, e*valuation(a)) + v)
setprecision!(Qp, e*precision(a)+ Int(_valuation_integral(a)) + v)
bb = inv(Qp(b))
setprecision!(Qp, old)
return a*bb
Expand Down Expand Up @@ -1046,7 +1040,7 @@ function expansion(x::NonArchLocalFieldElem, pi = uniformizer(parent(x)))
K = parent(x)
F, KtoF = residue_field(K)
Rt, t = laurent_series_ring(F, prec, :x)
v = ZZ(valuation(x) * absolute_ramification_index(K))
v = _valuation_integral(x)
y = divexact(x, pi^v)
coeffs = elem_type(F)[]
res = zero(Rt)
Expand Down
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