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CompactRepresentation.jl
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CompactRepresentation.jl
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@doc raw"""
compact_presentation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int = 2; decom, arb_prec = 100, short_prec = 1000) -> FacElem
Computes a presentation $a = \prod a_i^{n_i}$ where all the exponents $n_i$ are powers of $n$
and, the elements $a_i$ are "small", generically, they have a norm bounded by $d^{n/2}$ where
$d$ is the discriminant of the maximal order.
As the algorithm needs the factorisation of the principal ideal generated by $a$, it can
be passed in in \code{decom}.
"""
function compact_presentation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, nn::Int = 2; decom=false, arb_prec = 100, short_prec = 128)
n = ZZRingElem(nn)
K = base_ring(a)
if isempty(a.fac)
return a
end
if typeof(decom) == Bool
ZK = lll(maximal_order(K))
de::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem} = factor_coprime(IdealSet(ZK), a, refine = true)
else
#de = Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}((p, v) for (p, v) = decom)
de = Dict((p, v) for (p, v) = decom)
if length(decom) == 0
ZK = lll(maximal_order(K))
else
ZK = order(first(keys(decom)))
end
end
de_inv = Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsSimpleNumFieldOrderFractionalIdeal}()
be = FacElem(K(1))
if !(decom isa Bool)
@hassert :CompactPresentation 1 (length(de) == 0 && isone(abs(factored_norm(a)))) ||
(abs(factored_norm(a)) == factored_norm(FacElem(de)))
end
v = conjugates_arb_log_normalise(a, arb_prec)
if length(de) == 0
_v = FlintZZ(1)
else
_v = maximum(abs, values(de))+1
end
#Step 1: reduce the ideal in a p-power way...
A = ideal(ZK, 1)
@vprintln :CompactPresentation 1 "First reduction step"
cached_red = Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Dict{Int, Tuple{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}}}}()
n_iterations = Int(flog(_v, n))
for _k = n_iterations:-1:0
@vprintln :CompactPresentation 3 "Reducing the support: step $(_k) / $(n_iterations)"
B = Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}()
for (p, vv) in de
e_p = Int(div(vv, n^_k) % nn)
if iszero(e_p)
continue
end
if haskey(cached_red, p)
Dp = cached_red[p]
if haskey(Dp, e_p)
Ap, ap = Dp[e_p]
else
Ap, ap = power_reduce(p, ZZRingElem(e_p))
Dp[e_p] = (Ap, ap)
end
add_to_key!(B, Ap, 1)
mul!(be, be, ap^(-(n^_k)))
v -= Ref(n^_k) .* conjugates_arb_log_normalise(ap, arb_prec)
else
Dp = Dict{Int, Tuple{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}}}()
Ap, ap = power_reduce(p, ZZRingElem(e_p))
Dp[e_p] = (Ap, ap)
cached_red[p] = Dp
add_to_key!(B, Ap, 1)
v -= Ref(n^_k) .* conjugates_arb_log_normalise(ap, arb_prec)
mul!(be, be, ap^(-(n^_k)))
end
end
add_to_key!(B, A, n)
@vtime :CompactPresentation 3 A, alpha = reduce_ideal(FacElem(B))
mul!(be, be, alpha^(-(n^_k)))
#be *= alpha^(-(n^_k))
v -= Ref(n^_k) .* conjugates_arb_log_normalise(alpha, arb_prec)
end
if length(be.fac) > 1
delete!(be.fac, K(1))
end
#Step 2: now reduce the infinite valuation
r1, r2 = signature(K)
if length(de) == 0
m = FlintZZ(1)
else
m = maximum(abs, values(de))
end
m = max(m, 1)
local mm
while true
try
mm = abs_upper_bound(ZZRingElem, log(1+maximum(abs, v))//log(n))
break
catch e
if !isa(e, InexactError)
rethrow(e)
end
arb_prec *= 2
@vprintln :CompactPresentation 2 "increasing precision to $arb_prec"
v = conjugates_arb_log_normalise(a, arb_prec) +
conjugates_arb_log_normalise(be, arb_prec)
end
end
k = max(ceil(Int, log(m)/log(n)), Int(mm))
de = Dict(A => ZZRingElem(1))
delete!(de, ideal(ZK, 1))
@hassert :CompactPresentation 1 length(de) == 0 && isone(abs(factored_norm(a*be))) == 1 ||
abs(factored_norm(a*be)) == factored_norm(FacElem(de))
@hassert :CompactPresentation 2 length(de) != 0 || isone(ideal(ZK, a*be))
@hassert :CompactPresentation 2 length(de) == 0 || ideal(ZK, a*be) == FacElem(de)
while k>=1
@vprintln :CompactPresentation 1 "k now: $k"
D = Dict((p, div(ZZRingElem(v), n^k)) for (p, v) = de if v >= n^k)
if length(D) == 0
A = FacElem(Dict(ideal(ZK, 1) => 1))
else
A = FacElem(D)
end
vv = ArbFieldElem[x//n^k for x = v]
vvv = ZZRingElem[]
el_embs = a*be
for i=1:r1
while !radiuslttwopower(vv[i], -5)
arb_prec *= 2
v = conjugates_arb_log_normalise(el_embs, arb_prec)
vv = ArbFieldElem[x//n^k for x = v]
end
push!(vvv, round(ZZRingElem, vv[i]//log(2)))
end
for i=r1+1:r1+r2
while !radiuslttwopower(vv[i], -5)
arb_prec *= 2
v = conjugates_arb_log_normalise(el_embs, arb_prec)
vv = ArbFieldElem[x//n^k for x = v]
end
local r = round(ZZRingElem, vv[i]//log(2)//2)
push!(vvv, r)
push!(vvv, r)
end
@assert abs(sum(vvv)) <= degree(K)
@vtime :CompactPresentation 1 eA = (simplify(evaluate(A, coprime = true)))
@vtime :CompactPresentation 1 id = inv(eA)
local b
while true
@vtime :CompactPresentation 1 b = short_elem(id, matrix(FlintZZ, 1, length(vvv), vvv), prec = short_prec) # the precision needs to be done properly...
if abs(norm(b)//norm(id))> ZZRingElem(2)^abs(sum(vvv))*ZZRingElem(2)^degree(K)*abs(discriminant(ZK)) # the trivial case
short_prec *= 2
continue
else
break
end
end
@assert abs(norm(b)//norm(id)) <= ZZRingElem(2)^abs(sum(vvv))*ZZRingElem(2)^degree(K)* abs(discriminant(ZK)) # the trivial case
for (p, v) in A
if isone(p)
continue
end
de[p] -= n^k*v
end
@vtime :CompactPresentation 1 B = simplify(b*eA)
@assert isone(B.den)
B1 = B.num
@assert norm(B1) <= abs(discriminant(ZK))
@vprintln :CompactPresentation 1 "Factoring ($(B1.gen_one), $(B1.gen_two)) of norm $(norm(B1))"
@vtime :CompactPresentation 1 lfB1 = factor_easy(B1)
for (p, _v) = lfB1
if haskey(de, p)
de[p] += _v*n^k
elseif is_prime_known(p) && is_prime(p)
insert_prime_into_coprime!(de, p, _v*n^k)
else
de = insert_into_coprime(de, p, _v*n^k)
end
end
v_b = conjugates_arb_log_normalise(b, arb_prec)
@v_do :CompactPresentation 2 @show old_n = sum(x^2 for x = v)
v += Ref(n^k) .* v_b
@v_do :CompactPresentation 2 @show new_n = sum(x^2 for x = v)
@v_do :CompactPresentation 2 @show old_n / new_n
add_to_key!(be.fac, b, n^k)
#be *= FacElem(b)^(n^k)
@hassert :CompactPresentation 1 length(de) == 0 && isone(abs(factored_norm(a*be))) == 1 ||
abs(factored_norm(a*be)) == factored_norm(FacElem(de))
@hassert :CompactPresentation 2 length(de) != 0 || isone(ideal(ZK, a*be))
@hassert :CompactPresentation 2 length(de) == 0 || ideal(ZK, a*be) == FacElem(de)
k -= 1
end
if isempty(de)
de[ideal(ZK, 1)] = 1
end
@hassert :CompactPresentation 2 length(de) != 0 || isone(ideal(ZK, a*be))
@hassert :CompactPresentation 2 length(de) == 0 || ideal(ZK, a*be) == FacElem(de)
@hassert :CompactPresentation 1 length(de) == 0 && isone(abs(factored_norm(a*be))) == 1 ||
factored_norm(ideal(ZK, a*be)) == abs(factored_norm(FacElem(de)))
@vprintln :CompactPresentation 1 "Final eval..."
@vtime :CompactPresentation 1 A = evaluate(FacElem(de), coprime = true)
@vtime :CompactPresentation 1 b_ev = evaluate_mod(a*be, A)
inv!(be)
add_to_key!(be.fac, b_ev, ZZRingElem(1))
return be
end
function insert_prime_into_coprime!(de::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, e::ZZRingElem)
@assert !isone(p)
P = p.gen_one
for (k, v) in de
if k.gen_one % P == 0
if k.splitting_type[2] == 0
#k is not known to be prime, so p could divide...
v1 = valuation(k, p)
if v1 == 0
continue
end
#since it divides k it cannot divide any other (coprime!)
p2 = simplify(k*inv(p)^v1).num
if !isone(p2)
de[p2] = v
end
de[p] = v*v1+e
delete!(de, k)
return nothing
else
#both are know to be prime
@assert is_prime_known(k) && is_prime(k)
if k == p
# if they are equal
de[p] = v + e
return nothing
end
end
end
end
de[p] = e
return nothing
end
function insert_into_coprime(de::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, e::ZZRingElem)
@assert !isone(p)
P = p.gen_one
cp = AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}[]
for (k, v) in de
if !is_coprime(k.gen_one, P)
push!(cp, k)
end
end
if isempty(cp)
de[p] = e
return de
end
push!(cp, p)
cp = coprime_base(cp)
de1 = Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}()
for (k, v) in de
if is_coprime(k.gen_one, P)
de1[k] = v
end
end
for pp in cp
vp = e*valuation(p, pp)
for (k, v) in de
vp += valuation(k, pp)*v
end
if !iszero(vp)
de1[pp] = vp
end
end
return de1
end
#TODO: use the log as a stopping condition as well
@doc raw"""
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
Evaluates $a$ using CRT and small primes. Assumes that the ideal generated by $a$ is in fact $B$.
Useful in cases where $a$ has huge exponents, but the evaluated element is actually "small".
"""
function evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal)
K = base_ring(a)
if isempty(a.fac)
return one(K)
end
p = ZZRingElem(next_prime(p_start))
# p = ZZRingElem(next_prime(10000))
ZK = order(B)
dB = denominator(B)#*denominator(basis_matrix(ZK, copy = false))
@hassert :CompactPresentation 1 factored_norm(B) == abs(factored_norm(a))
@hassert :CompactPresentation 2 B == ideal(order(B), a)
@assert order(B) === ZK
pp = ZZRingElem(1)
re = K(0)
rf = ZK()
threshold = 3
if degree(K) > 30
threshold = div(degree(K), 10)
end
while (true)
dt = prime_decomposition_type(ZK, Int(p))
fl = true
for i = 1:length(dt)
if dt[i][1] > threshold
fl = false
break
end
end
if !fl
p = next_prime(p)
continue
end
local mp, me
try
me = modular_init(K, p)
mp = Ref(dB) .* modular_proj(a, me)
catch e
if !isa(e, BadPrime) && !isa(e, DivideError)
rethrow(e)
end
@show "badPrime", p
p = next_prime(p)
continue
end
m = modular_lift(mp, me)
if isone(pp)
re = m
rf = mod_sym(ZK(re), p)
pp = p
else
p2 = pp*p
last = rf
re = induce_inner_crt(re, m, pp*invmod(pp, p), p2, div(p2, 2))
rf = mod_sym(ZK(re), p2)
if rf == last
return nf(ZK)(rf)//dB
end
pp = p2
end
@hassert :CompactPresentation 1 nbits(pp) < 10000
p = next_prime(p)
end
end
function Hecke.is_power(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int; with_roots_unity = false, decom = false, trager = false, easy = false)
if n == 1
return true, a
end
@assert n > 1
K = base_ring(a)
if isempty(a.fac)
return true, FacElem(one(K))
end
anew_fac = Dict{AbsSimpleNumFieldElem, ZZRingElem}()
rt = Dict{AbsSimpleNumFieldElem, ZZRingElem}()
for (k, v) in a
if iszero(v)
continue
end
if isone(k)
continue
end
q, r = divrem(v, n)
if easy
if r <= 0
q -= 1
r += n
end
end
if !iszero(q)
rt[k] = q
end
if !iszero(r)
anew_fac[k] = r
end
end
if isempty(anew_fac)
K = base_ring(a)
if isempty(rt)
rt[one(K)] = 1
end
return true, FacElem(K, rt)
end
anew = FacElem(K, anew_fac)
if easy
fl, res1 = is_power(evaluate(anew), n, with_roots_unity = with_roots_unity)
res = FacElem(K, rt)*res1
return fl, res
end
c = conjugates_arb_log(a, 64)
c1 = conjugates_arb_log(anew, 64)
b = maximum(ZZRingElem[upper_bound(ZZRingElem, abs(x)) for x in c])
b1 = maximum(ZZRingElem[upper_bound(ZZRingElem, abs(x)) for x in c1])
if b1 <= isqrt(b)
fl, res = _ispower(anew, n, with_roots_unity = with_roots_unity, trager = trager)
if !fl
return fl, res
end
if !isempty(rt)
res = FacElem(rt)*res
end
return true, res
return r
end
return _ispower(a, n, with_roots_unity = with_roots_unity, decom = decom, trager = trager)
end
function _ispower(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int; with_roots_unity = false, decom = false, trager = false)
K = base_ring(a)
ZK = maximal_order(K)
@vprintln :Saturate 1 "Computing compact presentation"
@vtime :Saturate 1 c = Hecke.compact_presentation(a, n, decom = decom)
b = one(K)
d = Dict{AbsSimpleNumFieldElem, ZZRingElem}()
for (k, v) = c.fac
q, r = divrem(v, n)
if r < 0
r += n
q -= 1
@assert r > 0
@assert n*q+r == v
end
d[k] = q
mul!(b, b, k^r)
end
if isempty(d)
d[one(K)] = ZZRingElem(1)
end
df = FacElem(d)
@hassert :CompactPresentation 2 evaluate(df^n*b *inv(a))== 1
den = denominator(b, ZK)
fl, den1 = is_power(den, n)
if fl
den = den1
end
fl, x = is_power((den^n)*b, n, with_roots_unity = with_roots_unity, is_integral = true, trager = trager)
if fl
@hassert :CompactPresentation 2 x^n == b*(den^n)
add_to_key!(df.fac, K(den), -1)
return fl, df*x
else
return fl, df
end
end