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"""Hirzebruch L-polynomials -- numberdb.org/T336.
This generator stores the homogeneous Hirzebruch L-polynomials
L_n(p1, p2, ...) = [degree n] prod_i sqrt(x_i) / tanh(sqrt(x_i)),
where pj is the j-th elementary symmetric polynomial in the Pontryagin roots
x_i. The table starts at n = 1 and stops at n = 6, because L_7 has a nonzero
p7 term and the database searches polynomials in at most six variables.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
On the NumberDB build machine, where arguments are not passed through
agents/sage.sh, publish with:
$ cat "$NUMBERDB_KEY_FILE" | NUMBERDB_KEY_FROM_STDIN=1 NUMBERDB_PUBLISH=1 \
agents/sage.sh generate.py
The values are exact rational polynomials. The integrity checks compare the
first three nonconstant components with standard printed formulas, verify the
Pontryagin-root definition in seven roots, and check that substituting the
Pontryagin classes of T CP^(2m) gives signature 1 for 1 <= m <= 6.
"""
import os
import sys
from itertools import combinations
import numberdb.sage as numberdb
from sage.arith.misc import binomial
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
TABLE = os.environ.get("NUMBERDB_TABLE", "T336")
MAX_DEGREE = 6
CHECK_DEGREE = 7
OUTPUT_RING = PolynomialRing(
QQ, ["p%s" % i for i in range(1, MAX_DEGREE + 1)]
)
OUTPUT_P = OUTPUT_RING.gens()
CHECK_RING = PolynomialRing(
QQ, ["p%s" % i for i in range(1, CHECK_DEGREE + 1)]
)
CHECK_P = CHECK_RING.gens()
ROOT_RING = PolynomialRing(
QQ, ["x%s" % i for i in range(1, CHECK_DEGREE + 1)]
)
ROOTS = ROOT_RING.gens()
H_RING = PolynomialRing(QQ, "h")
H = H_RING.gen()
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
numberdb.configure(api_key=token)
def factorial(n):
value = QQ(1)
for k in range(2, int(n) + 1):
value *= QQ(k)
return value
def bernoulli_plus(n):
"""Bernoulli number with B_1 = +1/2. Only even values are used."""
values = [QQ(0) for _ in range(int(n) + 1)]
for m in range(int(n) + 1):
values[m] = QQ(1) / QQ(m + 1)
for j in range(m, 0, -1):
values[j - 1] = QQ(j) * (values[j - 1] - values[j])
return values[0]
def series_multiply(left, right, degree):
product = [QQ(0) for _ in range(degree + 1)]
for i, a in enumerate(left):
if a == 0:
continue
for j, b in enumerate(right):
if i + j > degree:
break
product[i + j] += QQ(a) * QQ(b)
return product
def l_series_coefficients(degree):
"""Coefficients of sqrt(x) / tanh(sqrt(x)) through x^degree."""
return [
(QQ(2) ** (2 * k)) * bernoulli_plus(2 * k) / factorial(2 * k)
for k in range(int(degree) + 1)
]
def log_series(series, degree):
"""Formal log of a series whose constant coefficient is 1."""
if series[0] != 1:
raise ValueError("log_series needs constant coefficient 1")
u = list(series)
u[0] -= QQ(1)
power = [QQ(1)] + [QQ(0) for _ in range(degree)]
out = [QQ(0) for _ in range(degree + 1)]
for exponent in range(1, degree + 1):
power = series_multiply(power, u, degree)
sign = QQ(1) if exponent % 2 else -QQ(1)
scale = sign / QQ(exponent)
for k in range(degree + 1):
out[k] += scale * power[k]
return out
def weighted_degree(exponents):
return sum((index + 1) * exponent for index, exponent in enumerate(exponents))
def ordinary_degree(exponents):
return sum(exponents)
def monomial_from_exponents(ring, variables, exponents):
monomial = ring.one()
for variable, exponent in zip(variables, exponents):
if exponent:
monomial *= variable ** exponent
return monomial
def truncate_weighted(polynomial, degree):
ring = polynomial.parent()
variables = ring.gens()
total = ring.zero()
for exponents, coefficient in polynomial.dict().items():
if weighted_degree(exponents) <= degree:
total += coefficient * monomial_from_exponents(
ring, variables, exponents
)
return total
def truncate_ordinary(polynomial, degree):
ring = polynomial.parent()
variables = ring.gens()
total = ring.zero()
for exponents, coefficient in polynomial.dict().items():
if ordinary_degree(exponents) <= degree:
total += coefficient * monomial_from_exponents(
ring, variables, exponents
)
return total
def homogeneous_weighted(polynomial, degree):
ring = polynomial.parent()
variables = ring.gens()
total = ring.zero()
for exponents, coefficient in polynomial.dict().items():
if weighted_degree(exponents) == degree:
total += coefficient * monomial_from_exponents(
ring, variables, exponents
)
return total
def power_sums(ring, variables, degree):
"""Power sums in terms of elementary symmetric polynomials."""
sums = [ring.zero()]
for m in range(1, int(degree) + 1):
total = ring.zero()
for j in range(1, m):
sign = QQ(1) if j % 2 else -QQ(1)
total += sign * variables[j - 1] * sums[m - j]
sign = QQ(1) if (m + 1) % 2 == 0 else -QQ(1)
total += sign * QQ(m) * variables[m - 1]
sums.append(truncate_weighted(total, degree))
return sums
def total_l_polynomial():
log_q = log_series(l_series_coefficients(CHECK_DEGREE), CHECK_DEGREE)
p = power_sums(CHECK_RING, CHECK_P, CHECK_DEGREE)
exponent = CHECK_RING.zero()
for degree in range(1, CHECK_DEGREE + 1):
exponent += log_q[degree] * p[degree]
exponent = truncate_weighted(exponent, CHECK_DEGREE)
total = CHECK_RING.one()
power = CHECK_RING.one()
for k in range(1, CHECK_DEGREE + 1):
power = truncate_weighted(power * exponent, CHECK_DEGREE)
total = truncate_weighted(total + power / factorial(k), CHECK_DEGREE)
return total
TOTAL_L = total_l_polynomial()
def to_output_ring(polynomial):
total = OUTPUT_RING.zero()
for exponents, coefficient in polynomial.dict().items():
if any(exponents[MAX_DEGREE:]):
raise AssertionError("term beyond p6 survived: %s" % (polynomial,))
monomial = OUTPUT_RING.one()
for variable, exponent in zip(OUTPUT_P, exponents[:MAX_DEGREE]):
if exponent:
monomial *= variable ** exponent
total += coefficient * monomial
return total
def l_polynomial(n):
return to_output_ring(homogeneous_weighted(TOTAL_L, int(n)))
def root_q(root):
coeffs = l_series_coefficients(CHECK_DEGREE)
return sum(coeffs[k] * root ** k for k in range(CHECK_DEGREE + 1))
def elementary_roots(k):
if k == 0:
return ROOT_RING.one()
total = ROOT_RING.zero()
for indexes in combinations(range(CHECK_DEGREE), int(k)):
monomial = ROOT_RING.one()
for index in indexes:
monomial *= ROOTS[index]
total += monomial
return total
ROOT_ELEMENTARY = [elementary_roots(k) for k in range(1, CHECK_DEGREE + 1)]
def root_product_l():
total = ROOT_RING.one()
for root in ROOTS:
total = truncate_ordinary(total * root_q(root), CHECK_DEGREE)
return total
def substitute_check_to_roots(polynomial):
return truncate_ordinary(ROOT_RING(polynomial(*ROOT_ELEMENTARY)), CHECK_DEGREE)
def complex_projective_space_substitution(polynomial, m):
"""Substitute p_j(T CP^(2m)) = binomial(2m + 1, j) h^j."""
images = [
binomial(2 * int(m) + 1, j) * H ** j
for j in range(1, MAX_DEGREE + 1)
]
return H_RING(polynomial(*images))
def iter_parameters():
for n in range(1, MAX_DEGREE + 1):
yield {"n": ZZ(n)}
def check_identities():
expected = {
1: OUTPUT_P[0] / QQ(3),
2: (7 * OUTPUT_P[1] - OUTPUT_P[0] ** 2) / QQ(45),
3: (
62 * OUTPUT_P[2]
- 13 * OUTPUT_P[0] * OUTPUT_P[1]
+ 2 * OUTPUT_P[0] ** 3
) / QQ(945),
}
for n, value in expected.items():
found = l_polynomial(n)
if found != value:
raise AssertionError("L_%s printed formula failed: %s != %s"
% (n, found, value))
l7 = homogeneous_weighted(TOTAL_L, 7)
p7_coefficient = l7.monomial_coefficient(CHECK_P[6])
if p7_coefficient == 0:
raise AssertionError("L_7 unexpectedly has no p7 term")
via_roots = root_product_l()
via_components = substitute_check_to_roots(TOTAL_L)
if via_components != via_roots:
raise AssertionError("root-product check failed")
for m in range(1, MAX_DEGREE + 1):
value = complex_projective_space_substitution(l_polynomial(m), m)
if value[m] != 1:
raise AssertionError(
"signature of CP^(2*%s) failed: coefficient is %s"
% (m, value[m])
)
lengths = [
(len(str(l_polynomial(n))), n)
for n in range(1, MAX_DEGREE + 1)
]
longest = max(lengths, key=lambda item: item[0])
print("integrity checks passed for %d Hirzebruch L-polynomials"
% MAX_DEGREE)
print("matched the printed formulas for L_1 through L_3")
print("root-product and projective-space signature checks passed")
print("L_7 has p7 coefficient %s" % p7_coefficient)
print("longest polynomial has %d characters at n=%s"
% (longest[0], longest[1]))
class HirzebruchLPolynomials(numberdb.Generator):
table = TABLE
parameters = ("n",)
type = "Q[]"
rigour = "exact"
def enumerate(self):
yield from iter_parameters()
def value(self, params, digits):
return l_polynomial(ZZ(params["n"]))
def fill_draft_once(generator, message):
"""Fill a fresh prose draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message, rigour=generator.rigour)
return answer
def stored_values():
table = numberdb.table(TABLE)
numbers = table.get("Numbers") or {}
values = {}
if isinstance(numbers, dict):
for key, value in numbers.items():
if isinstance(value, dict):
value = value.get("number")
values[ZZ(key)] = OUTPUT_RING(str(value))
elif isinstance(numbers, list):
for row in numbers:
params = row.get("params", row)
value = row.get("number")
values[ZZ(params["n"])] = OUTPUT_RING(str(value))
return values
def check_stored_values():
values = stored_values()
if len(values) != MAX_DEGREE:
raise AssertionError("stored table has %s values, expected %s"
% (len(values), MAX_DEGREE))
for n in range(1, MAX_DEGREE + 1):
expected = l_polynomial(n)
if values.get(ZZ(n)) != expected:
raise AssertionError("stored L_%s failed: %s != %s"
% (n, values.get(ZZ(n)), expected))
print("stored values match the independently checked generator")
def main():
_key_from_stdin()
check_identities()
generator = HirzebruchLPolynomials()
if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
print(fill_draft_once(
generator,
message="fill Hirzebruch L-polynomials from exact characteristic series",
))
else:
report = generator.verify(sample=None)
print(report)
if not report.ok:
sys.exit(1)
check_stored_values()
if __name__ == "__main__":
main()