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11489 bytes, as of the version from 2026-09-20 05:29 (current). Recorded here, not run.
"""Sharp Gagliardo-Nirenberg constants of Del Pino and Dolbeault -- numberdb.org/T358.
This generator fills T358 with the sharp constants A_{n,a} in the
Del Pino-Dolbeault Gagliardo-Nirenberg inequality for 3 <= n <= 20 and
rational a of denominator at most 8 in the admissible open range, together
with the Sobolev endpoint a = n/(n-2).
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
"""
import os
import re
import sys
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
TABLE = os.environ.get("NUMBERDB_TABLE", "T358")
DIGITS = 100
WORKING_GUARD = 160
CHECK_GUARD = 256
MAX_N = 20
MAX_DENOMINATOR = 8
SOBOLEV_TABLE = "T92"
FRACTIONAL_SOBOLEV_TABLE = "T354"
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 _real_field(digits, guard=WORKING_GUARD):
return RealBallField(numberdb.bits(digits, losing=guard))
def _theta(n, a):
n = QQ(n)
a = QQ(a)
return n * (a - 1) / (a * (n + 2 - (n - 2) * a))
def _endpoint(n):
return QQ(n) / QQ(n - 2)
def _admissible_exponents(n):
endpoint = _endpoint(n)
values = {endpoint}
for denominator in range(1, MAX_DENOMINATOR + 1):
for numerator in range(denominator + 1, int(endpoint * denominator) + 1):
a = QQ(numerator) / QQ(denominator)
if QQ(1) < a < endpoint and a.denominator() <= MAX_DENOMINATOR:
values.add(a)
return sorted(values)
def _constant(n, a, digits, guard=WORKING_GUARD):
field = _real_field(digits, guard)
n = QQ(n)
a = QQ(a)
y = (a + 1) / (a - 1)
theta = _theta(n, a)
n_ball = field(n)
a_ball = field(a)
y_ball = field(y)
theta_ball = field(theta)
return (
(y_ball * field((a - 1) ** 2) / (2 * field.pi() * n_ball)) ** (theta_ball / 2)
* ((2 * y_ball - n_ball) / (2 * y_ball)) ** (1 / (2 * a_ball))
* (y_ball.gamma() / field(y - n / 2).gamma()) ** (theta_ball / n_ball)
)
def _integral_power_norm(n, exponent, s, digits, guard=CHECK_GUARD):
field = _real_field(digits, guard)
n = QQ(n)
exponent = QQ(exponent)
s = QQ(s)
alpha = s * exponent
integral = (
field.pi() ** (field(n) / 2)
* field(alpha - n / 2).gamma()
/ field(alpha).gamma()
)
return integral ** (field(QQ(1) / exponent))
def _gradient_norm(n, s, digits, guard=CHECK_GUARD):
field = _real_field(digits, guard)
n = QQ(n)
s = QQ(s)
beta = 2 * s + 2
integral = (
2
* field(n)
* field(s) ** 2
* field.pi() ** (field(n) / 2)
* field(beta - n / 2 - 1).gamma()
/ field(beta).gamma()
)
return integral.sqrt()
def _extremal_ratio(n, a, digits, guard=CHECK_GUARD):
field = _real_field(digits, guard)
n = QQ(n)
a = QQ(a)
s = QQ(1) / (a - 1)
theta = _theta(n, a)
left = _integral_power_norm(n, 2 * a, s, digits, guard)
grad = _gradient_norm(n, s, digits, guard)
middle = _integral_power_norm(n, a + 1, s, digits, guard)
return left / (grad ** field(theta) * middle ** field(1 - theta))
def _sobolev_p2_values():
table = numberdb.table(SOBOLEV_TABLE)
values = {}
for n_text, by_p in (table.get("Numbers") or {}).items():
if "2" not in by_p:
continue
by_q = by_p["2"]
if len(by_q) != 1:
raise ArithmeticError("unexpected T92 p=2 row for n=%s" % n_text)
values[ZZ(n_text)] = next(iter(by_q.values()))
return values
def _fractional_sobolev_s1_values():
table = numberdb.table(FRACTIONAL_SOBOLEV_TABLE)
values = {}
for n_text, by_s in (table.get("Numbers") or {}).items():
if "1" in by_s:
values[ZZ(n_text)] = by_s["1"]
return values
def _stored_decimal_ball(text, digits, guard=CHECK_GUARD):
field = _real_field(digits, guard)
if isinstance(text, dict):
text = text["number"]
value = field(str(text))
found = re.fullmatch(r"-?\d+\.(\d+)(?:[eE](-?\d+))?", str(text))
if not found:
return value
places = len(found.group(1)) - int(found.group(2) or 0)
return value.add_error(field(10) ** (-places))
def _entry_number(entry):
if isinstance(entry, dict):
return entry.get("number")
return entry
class GagliardoNirenbergDelPinoDolbeaultConstants(numberdb.Generator):
"""Generator for T358, the Del Pino-Dolbeault Gagliardo-Nirenberg constants."""
table = TABLE
parameters = ("n", "a")
type = "R"
digits = DIGITS
rigour = "proven"
def enumerate(self, max_n=MAX_N):
for n in range(3, max_n + 1):
for a in _admissible_exponents(n):
yield {"n": str(n), "a": str(a)}
def value(self, params, digits):
n = ZZ(params["n"])
a = QQ(params["a"])
value = _constant(n, a, digits)
if a == _endpoint(n):
return {
"number": value,
"comment": "Sobolev endpoint: this value is $1/S_{%s,2}$." % n,
}
return value
def run_integrity_checks():
field = _real_field(DIGITS, CHECK_GUARD)
generator = GagliardoNirenbergDelPinoDolbeaultConstants()
values = {}
widest_radius = field(0)
widest_at = None
for params in generator.enumerate():
n = ZZ(params["n"])
a = QQ(params["a"])
value = _constant(n, a, DIGITS, CHECK_GUARD)
values[(n, a)] = value
ratio = _extremal_ratio(n, a, DIGITS, CHECK_GUARD)
if not (value - ratio).contains_zero():
raise ArithmeticError(
"extremal ratio check failed at n=%s, a=%s: %s vs %s"
% (n, a, value, ratio)
)
radius = field(value.rad())
if radius > widest_radius:
widest_radius = radius
widest_at = (n, a)
sobolev = _sobolev_p2_values()
fractional = _fractional_sobolev_s1_values()
endpoint_checked = []
t92_checked = []
for n in range(3, MAX_N + 1):
if ZZ(n) not in fractional:
raise ArithmeticError("T354 has no s=1 row for n=%s" % n)
a = _endpoint(n)
product = values[(ZZ(n), a)] * _stored_decimal_ball(fractional[ZZ(n)], DIGITS)
if not (product - 1).contains_zero():
raise ArithmeticError(
"fractional Sobolev endpoint check failed at n=%s: product %s"
% (n, product)
)
endpoint_checked.append(n)
if ZZ(n) in sobolev:
product_t92 = values[(ZZ(n), a)] * _stored_decimal_ball(sobolev[ZZ(n)], DIGITS)
if not (product_t92 - 1).contains_zero():
raise ArithmeticError(
"T92 endpoint check failed at n=%s: product %s" % (n, product_t92)
)
t92_checked.append(n)
print("integrity checks passed for %d entries" % len(values))
print("widest value ball radius: %s at n=%s, a=%s" % (
widest_radius, widest_at[0], widest_at[1]))
print("Sobolev endpoints checked against %s for n=%s" % (
FRACTIONAL_SOBOLEV_TABLE, ",".join(str(n) for n in endpoint_checked)))
print("Sobolev endpoints also checked against %s for n=%s" % (
SOBOLEV_TABLE, ",".join(str(n) for n in t92_checked)))
def run_stored_checks():
table = numberdb.table(TABLE)
numbers = table.get("Numbers") or {}
if not numbers:
print("stored checks skipped: %s has no stored numbers yet" % TABLE)
return
checked = 0
endpoint_checked = 0
sobolev = _sobolev_p2_values()
fractional = _fractional_sobolev_s1_values()
for n_text, by_a in numbers.items():
n = ZZ(n_text)
for a_text, entry in by_a.items():
a = QQ(a_text)
stored = _stored_decimal_ball(_entry_number(entry), DIGITS)
ratio = _extremal_ratio(n, a, DIGITS, CHECK_GUARD)
if not (stored - ratio).contains_zero():
raise ArithmeticError(
"stored extremal check failed at n=%s, a=%s: %s vs %s"
% (n, a, stored, ratio)
)
checked += 1
if a == _endpoint(n):
product = stored * _stored_decimal_ball(fractional[n], DIGITS)
if not (product - 1).contains_zero():
raise ArithmeticError(
"stored fractional endpoint check failed at n=%s: product %s"
% (n, product)
)
if n in sobolev:
product_t92 = stored * _stored_decimal_ball(sobolev[n], DIGITS)
if not (product_t92 - 1).contains_zero():
raise ArithmeticError(
"stored T92 endpoint check failed at n=%s: product %s"
% (n, product_t92)
)
endpoint_checked += 1
print("stored checks passed for %d entries and %d endpoints" % (
checked, endpoint_checked))
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
if __name__ == "__main__":
_key_from_stdin()
generator = GagliardoNirenbergDelPinoDolbeaultConstants()
run_integrity_checks()
if "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1":
print(fill_draft_once(
generator,
message="Del Pino-Dolbeault Gagliardo-Nirenberg constants"))
else:
run_stored_checks()
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)