New computations of Poisson polynomials
David H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley National Lab (retired)
University of California, Davis

January 11, 2023

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Polynomials arising from the Poisson potential function
The Poisson potential function appears in numerous applied math contexts, ranging
from mathematical physics to sharpening iPhone images:
1 X cos(mπx ) cos(nπy )φ2 (x , y ) = 2
π m,n odd
m2 + n2
In a 2013 study, colleagues and I numerically discovered and then proved the intriguingfact that for rational (x , y ), the 2-D Poisson potential function φ2 (x , y ) satisfies
1
φ2 (x , y ) = log απwhere α is algebraic, i.e., the root of an integer polynomial of some degree m.
By computing high-precision numerical values of φ2 (x , y ) for various specific rational x
and y , and applying variants of the PSLQ program, we were able to produce theexplicit minimal polynomials for α in several simple specific cases.
I D. H. Bailey, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arising from the Poisson
equation,” Journal of Physics A: Mathematical and Theoretical, vol. 46 (2013), 115201.

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The PSLQ integer relation algorithm
Let X = (xk ) be an (m + 1)-long real or complex vector. An integer relation algorithm
finds a nontrivial integer vector A = (ak ) such thata0 x0 + a1 x1 + · · · + am xm = 0.
I The PSLQ algorithm and the multipair PSLQ algorithm are commonly used for
integer relation detection. Variants of the LLL algorithm can also be used.
I Integer relation detection (by any algorithm) requires very high precision: at least(m + 1) · maxk log10 |ak | digits, both in the input data and the algorithm.
I Fast variations of PSLQ and multipair PSLQ utilize two, three or even four levelsof precision, doing as much computation as possible with only double precision,and switching to higher levels only when needed.

I D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques andapplications,” Mathematics of Computation, vol. 70, no. 236 (Oct 2000), 1719–1736.
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Numerically finding minimal polynomials using PSLQPSLQ can be used to recognize a computed numerical value as the root of an integer
polynomial of degree m. Example: The following constant is suspected to be an
algebraic number:
α = 1.232688913061443445331472869611255647068988824547930576057634684778 . . .
What is its minimal polynomial?
Method: Compute the vector (1, α, α2 , · · · , αm ) for m = 30, then input this vector to
PSLQ or some other integer relation algorithm.
Answer (using 250-digit arithmetic):
0 = 697 − 1440α − 20520α2 − 98280α3 − 102060α4 − 1458α5 + 80α6 − 43920α7+ 538380α8 − 336420α9 + 1215α10 − 80α12 − 56160α13 − 135540α14 − 540α15+ 40α18 − 7380α19 + 135α20 − 10α24 − 18α25 + α30
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Some initial Poisson polynomials found using PSLQ
s
5
6
7
8
9

10

Minimal polynomial corresponding to x = y = 1/s:
1 + 52α − 26α2 − 12α3 + α4
1 − 28α + 6α2 − 28α3 + α4
−1 − 196α + 1302α2 − 14756α3 + 15673α4 + 42168α5 − 111916α6 + 82264α7−35231α8 + 19852α9 − 2954α10 − 308α11 + 7α12
1 − 88α + 92α2 − 872α3 + 1990α4 − 872α5 + 92α6 − 88α7 + α8
−1 − 534α + 10923α2 − 342864α3 + 2304684α4 − 7820712α5 + 13729068α6−22321584α7 + 39775986α8 − 44431044α9 + 19899882α10 + 3546576α11−8458020α12 + 4009176α13 − 273348α14 + 121392α15−11385α16 − 342α17 + 3α18
1 − 216α + 860α2 − 744α3 + 454α4 − 744α5 + 860α6 − 216α7 + α8These computations were very expensive. The case x = y = 1/32, for instance,
required 10,000-digit arithmetic and ran for 45 hours. Some larger runs, using evenhigher precision, failed.
Help! More powerful computational tools are required.
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Kimberley’s formula for the degree of the polynomial
Based on these preliminary results, Jason Kimberley of the University of Newcastle,
Australia observed that the degree m(s) of the minimal polynomial associated with thecase x = y = 1/s appears to be given by the following:
Set m(2) = 1/2. Otherwise for primes p congruent to 1 mod 4, set m(p) = int2 (p/2),
where int denotes greatest integer, and for primes p congruent to 3 mod 4, setm(p) = int (p/2)(int (p/2) + 1). Then for any other positive integer s whose prime
factorization is s = p1e1 p2e2 · · · prer ,r −1m(s) = 4rY
2(ei −1)pim(pi ).

i=1

Does Kimberley’s formula hold for larger s?
What is the true mathematical connection between the pair of rationals (x , y ) and thealgebraic number α?
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2015: Improvements to the Poisson polynomial computation program
1. A new multiprecision package: 3X faster than before, suitable for use in a
multi-threaded parallel environment.
2. A new 3-level multipair PSLQ program, up to 5X faster:a Double precision (appr. 15 digit accuracy).
b Medium multiprecision: varies from 100 digits to 1200 digits on Poisson problems.
c Full multiprecision: varies from 2,500 digits to 50,000 digits on Poisson problems.

3. Faster hardware: a new 8-core MacPro system with Intel processor.

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Key breakthrough: A fast algorithm to compute φ2 (x , y )The original formula for φ2 (x , y ) converges much too slowly for numerical evaluation.
But this formula, found by Jonathan Borwein, converges very rapidly:
1θ2 (z, q)θ4 (z, q)log
φ2 (x , y ) =,2π
θ1 (z, q)θ3 (z, q)
where q = e −π and z = π2 (y + ix ), and whereθ1 (z, q) = 2
θ2 (z, q) = 2∞
X2(−1)k−1 q (2k−1) /4 sin((2k − 1)z),

k=1
∞
X

2

q (2k−1) /4 cos((2k − 1)z),k=1

θ3 (z, q) = 1 + 2
θ4 (z, q) = 1 + 2

∞
X
k=1
∞
X

2

q k cos(2kz),
2(−1)k q k cos(2kz).

k=1
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High-level computational algorithm
1. Given rationals x = p/s and y = q/s, select a conjectured minimal polynomial
degree m(s) (using Kimberley’s formula) and other parameters for the run.
2. Calculate φ2 (x , y ) to P2 -digit precision using the formulas from the previous
viewgraph. When done, calculate α = exp(8πφ2 (x , y )) and generate the(m + 1)-long vector X = (1, α, α2 , · · · , αm ), to P2 -digit precision.
3. Apply the three-level multipair PSLQ algorithm to X .
4. If no numerically significant relation is found, try again with a larger degree m or
higher precision P2 . If a relation is found, employ the polynomial factorization
facilities in Mathematica and Maple to ensure that the polynomial is irreducible.

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192-degree minimal polynomial found by multipair PSLQ for x = y = 1/35
−1−127936α1−119729568α2+86259615424α3−63414593420752α4−23761356230294720α5−21521006149475920864α6+611939502140287079872α7−368241444434585386649704α8+34534137997535622208800832α9+190487328667825240861480928α10+70731866698900460686569086656α11−30639321248717091789168538879600α12+1922209242084456061370070376570688α13−92225937246234731111918315818791008α14+5237085972458845348795423777776516544α15−235187168740813424983476512954481139732α16+7429223399159195923652315325576416851776α17−183888805409443551177519325744878574722592α18+3271185405201792963719366162166966763587520α19−30299858585212592953207274837044089721625936α20−359992131877582010695076712447360628122541504α21+21102025339314733846086441916632636325491195040α22−476414747523697562008419279360098519444810134848α23+7169682338305684610208768903330082567075749930088α24−77147038334570163492295981291240215533152479746240α25+576164594448741453064385715717563185909190291963744α26−2736845608824484767533372475869861068967680432337984α27+14259726273826198867906144815986191950918103600120720α28−418898418536517746458185180042147268305073686192184768α29+11576727630588055309375415098495616370526328495842993184α30−215864800528660839585002162871715414654518484882221347136α31+2990075428515164844805618172618994337159659201108555443326α32−32103405822691698883721869657536188329165809065032597536320α33+266563559238616545234045835785848715794979563412588549450912α34−1588318769762074127932309766905276050093368042694667151097024α35+4203983024865006644081516487096952711041894782888827508128720α36+45741795572596990194281792252600295928845217432176402329342656α37−888295322846459251495779433538199608179522746791003115766467360α38+9318868074863097497435367536227145321575550333899160360356887616α39−75360568030730663716613169159158032198635749984420852664531432312α40+511904470165868318936871843389229757000581029769096726546021748160α41−3023546278817700158319248501909046229060364677626955289951128897760α42+15762439699463291788277556458704672972787957041647009977980115608384α43−72736254087237112123433235647299134668422519990232924303557659578512α44+294655977143150502457165877012542972911987116693855121083332478262976α45−1023418666617067140855492177628022043641503497647927192890895515496096α46+2872077837131433597858520147550093715024860106120898508958083768560704α47−5244656757384067630568130031166724732476128944651080774800327815250116α48−4375176099824059357126327245955265067024929918868045662842672259365696α49+106846263928061290720995820179791755195031418846948198250527939644422688α50−723639781157123885224054475278025279616831219096501715148796353565656000α51+3859388734169523941501510320228124865152568559427283813598867448036672464α52−18252524309522593867333335673661626604394606878814374964432136569382704704α53+77801314121940709085266334072671847312436136451041737155601244216316832608α54−295924487894698517430363594146438746974018698578472622597771256623091267264α55+990883094386701201740101120896676716687418747972363663196338999871139087736α56−2890647897108739627601089028825831836951153432467718962671967596342224331584α57+7329280206578701233923536747523603867159738577214393279268225554697992024224α58−16529926761499923756105727653427646107756611125773189972311741253023019544512α59+36621401257350141587688985844214102107297318891842289523223570545646432107248α60−97806177157646206090474078202543089426709440278475346399840243186928760712768α61+342578593578669077931375617617876157438332517966938868387799492392630391951328α62−1318494126999138496197980784859753020673610723076044612133782854663456341738176α63+4770648039063802379141989053619421884321933507524681278243277292208295535890193α64−15472388710953109453763060955215809455238864792962040591046439233238251454944640α65+44993126133327412718394996675533962755303579158603905782013384675151509701621696α66−118993678023270686331414029749780909390537125214711043931901528442137124714048640α67+291344068959030996949609862561213601556661749401236700666696596438800158224020448α68−672232544304160020140857460151229781783982323755372371953704546852887742475693952α69+1483756546102154371991621621590988762258290163435879224071206411473630010421819712α70−3160698702072288781771809020419553277725945343183442108817290538424291765729105536α71+6501409535318122589047227869235783252522832570490016457500770280864513874652673648α72−12819483008841743167047686498254296089188806357669061135254105501043843161878338944α73+23940036962388573943673383046497610566473660443396196928270109685668366770350448320α74−41783013070841149166516454269070382975961508352312427230333483156459522180289158272α75+67368007706334464267463678880196649459723255934263643619439168617608299192825564576α76−99642265418236291841000783588241107943576928164973329187609122090423042369846900608α77+135684188062925983721829145899247186881926375911313411751554577363782314702542167104α78−175027359690635875118927274192015444148378662148930449526142186378361018115616958080α79+23061047037097602217994243304224041474746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distinctive semi-elliptical pattern is strong evidence that the result is correct.
10 / 17

Selected runs (degrees, precision, timings, etc.) for x = y = 1/s
s
20
24
30
32
34
36
37
38
39
40
42
44
45
46
48
50

m
32
64
64
128
128
144
324
180
288
128
192
240
288
264
256
200

log10 (D)-463.84-1883.78-1868.01-7577.07-7574.93-9570.86-48431.32-14951.64-38330.14-7580.00-16993.99-26604.14-38315.08-32036.34-30248.55-18421.18

P1
160
320
350
650
650
750
1650
900
1450
650
1000
1200
1450
1350
1350
1000

P2
700
2200
2300
8200
8200
10300
51000
16000
40000
8200
18000
28000
40000
34000
32000
20000

N
1967
9297
9064
45893
45914
62282
93125412098466715345655
150364323762660001476902415316168947

M
0.81
11.33
11.33
168.20168.20267.106579.66642.984124.24
168.20829.41
2003.33
4124.24
2921.57
2586.39974.44

C
1
1
1
1
1
1
16
1
16
1
8
8
16
16
16
8

T (sec.)2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
4.84 · 105
3.88 · 104
2.68 · 105
5.02 · 103
1.57 · 104
7.43 · 104
2.09 · 105
1.06 · 105
8.98 · 104
2.12 · 104

C · T (sec.)
2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
7.74 · 106
3.88 · 104
4.29 · 106
5.02 · 103
1.26 · 105
5.94 · 105
3.35 · 106
1.70 · 106
1.44 · 106
1.69 · 105

s = denominator; m = degree; D = detection level; P1 = medium precision; P2 = full precision; N =number of iterations; M = Mbytes; C = cores; T = wall clock time; C · T = total core-seconds.
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Palindromic polynomials
From our results, in the case (1/s, 1/s) where s is even, the resulting polynomial isalways palindromic (ak = am−k ). For instance, when s = 16,
p16 (α) = 1 − 1376α1 − 12560α2 − 3550496α3 + 81241720α4 − 169589984α5+ 1334964944α6 − 24307725984α7 + 238934926108α8 − 1043027124704α9+ 2328675366384α10 − 3219896325280α11 + 4238551472456α12− 10247414430048α13 + 28552105805904α14 − 55832851687968α15+ 70020268309062α16− 55832851687968α17 + 28552105805904α18 − 10247414430048α19+ 4238551472456α20 − 3219896325280α21 + 2328675366384α22− 1043027124704α23 + 238934926108α24 − 24307725984α25 + 1334964944α26− 169589984α27 + 81241720α28 − 3550496α29 − 12560α30 − 1376α31 + α32

Nitya Mani, a student at Stanford University, pointed out that if α is a root of apalindromic polynomial such as this, then α + 1/α is a root of a transformed
polynomial of half the degree. This fact can be used to greatly accelerate thecomputation of Poisson polynomials when s is an even integer.

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2016: A proof of Kimberley formula for the case (1/s, 1/s)
Some observations from the polynomials produced by the program:
I The algebraic number αs is the largest real root of the associated polynomial.
I The polynomial has ϕ(s) real roots, where ϕ is the Euler totient function.
√
I −αs is the largest purely imaginary root of ψs (x , 1), where ψ is a sequence ofpolynomials defined in a 2010 paper by Savin and Quarfoot of the Univ. of Utah.
The Savin-Quarfoot paper was found by doing an Internet search for “387221579866,”
which is a coefficient of the polynomial for the case (1/11, 1/11).
These observations ultimately led to a proof, by Watson Ladd of U.C. Berkeley, ofKimberley’s formula in the case x = y = 1/s.
I D. H. Bailey, J. M. Borwein, J. Kimberley and W. Ladd, “Computer discovery and analysis oflarge Poisson polynomials,” Experimental Mathematics, 27 Aug 2016, vol. 26, 349–363,
https://www.davidhbailey.com/dhbpapers/poisson-res.pdf.

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What about the many cases with x 6= y ?
All of the computations and results mentioned above are for the case x = y = 1/s forsome positive integer s.
What about rationals x = p/s, y = q/s, with p 6= q? Initial results indicated that
Kimberley’s formula does not hold for these more general rationals.
Is there a generalization of Kimberley’s formula that holds in these cases? To
numerically address this more general problem requires many times more computation
than for the x = y = 1/s cases.
A familiar refrain: Help! More powerful computational tools are required.

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2022: New computational tools for the Poisson polynomial problem
1. A new multiprecision package, based on integer arithmetic, that is approximately
3X faster than before. As an option, this software can also use the MPFR and
GMP packages for low-level operations.
2. A new 4-level multipair PSLQ program, up to 1.5X faster:a Double precision (approx. 15 digit accuracy).
b Quad precision (approx. 32 digit accuracy).
c Medium multiprecision: varies from 100 digits to 1200 digits on Poisson problems.
d Full multiprecision: varies from 2,500 digits to 50,000 digits on Poisson problems.

3. The code does as much computation as possible in double precision; it shifts tohigher precision when double precision is exhausted.
4. Faster hardware: It is currently running on a 10-core MacStudio, with Apple’s new
low-power M1 Pro processor (thus helping the author avoid personal bankruptcy
from electric bills).

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December 2022: New computer runs
I The new software has been used to compute the minimal polynomials for theentire set of cases (p/s, q/s), where 1 ≤ p ≤ q < s/2, where s ranges from 10 to42 (except s = 37, 39, 41), and where gcd(p, q, s) = 1. A total of 2024 individual
cases have been processed.
I These runs required approximately four months run time on all 10 cores of thesystem, completing a few days ago.
Kimberley’s formula is not valid for these more general cases, but the following
modification of Kimberley’s formula appears to hold:1. For the cases x = y = 1/s, Kimberley’s formula holds.
2. For the cases x = p/s, y = q/s with s odd, Kimberley’s formula holds (except fora few cases when the correct degree is smaller by a power of two factor).
3. For the cases x = p/s, y = q/s, with s even and both p and q odd, Kimberley’s
formula holds (with a few exceptions as above).
4. For the cases x = p/s, y = q/s, with s even and one of p or q is even, the correct
degree is twice Kimberley’s formula (with a few exceptions as above).
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Conclusions
I New computations of Poisson polynomials have been completed, covering theentire set of cases (p/s, q/s), where 1 ≤ p ≤ q < s/2, where s ranges from 10 to42 (except s = 37, 39, 41), and where gcd(p, q, s) = 1. A total of 2024 individual
cases have been processed.
I The resulting minimal polynomials have shown that Kimberley’s formula does nothold for these more general cases. However, a modification of Kimberley’s formulaappears to hold.
I We currently have no idea how this modified rule can be proved.
I The three-dimensional and higher-dimensional Poisson formulas have not yet been
explored computationally; these cases likely will require even more computation.
I A familiar refrain: Help! More powerful computational tools are required.
This talk is available at
http://www.davidhbailey.com/dhbtalks/dhb-wcnt-2022.pdf.
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