Davenport-Stothers polynomial triples

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Polynomials

$M$

class

$f(t)$, $g(t)$, or $h(t)$

unique

$f(t)$:

4*t^2 + 1

unique

$g(t)$:

8*t^3 + 3*t

unique

$h(t)$:

3*t^2 + 1

unique

$f(t)$:

t^4 + 4*t

unique

$g(t)$:

t^6 + 6*t^3 + 6

unique

$h(t)$:

-8*t^3 - 36

unique

$f(t)$:

t^6 + 4*t^4 + 10*t^2 + 6

unique

$g(t)$:

t^9 + 6*t^7 + 21*t^5 + 35*t^3 + 63/2*t

unique

$h(t)$:

27*t^4 + 351/4*t^2 + 216

unique

$f(t)$:

t^8 - 2*t^7 + 7*t^6 - 6*t^5 + 11*t^4 + 4*t^3 + 12*t + 1

unique

$g(t)$:

t^12 - 3*t^11 + 12*t^10 - 19*t^9 + 39*t^8 - 24*t^7 + 30*t^6 + 36*t^5 - 15*t^4 + 60*t^3 + 27/2*t^2 + 9/2*t + 29/2

unique

$h(t)$:

-27*t^5 + 135/4*t^4 - 243/2*t^3 + 81/4*t^2 - 189/2*t - 837/4

Birch symmetric

$f(t)$:

1/9*t^10 + 2/3*t^7 + 5/3*t^4 + 4/3*t

Birch symmetric

$g(t)$:

1/27*t^15 + 1/3*t^12 + 4/3*t^9 + 8/3*t^6 + 5/2*t^3 + 1/2

Birch symmetric

$h(t)$:

-1/36*t^6 - 7/54*t^3 - 1/4

Elkies primitive

$f(t)$:

t^10 + 2*t^9 + 33*t^8 + 12*t^7 + 378*t^6 - 336*t^5 + 2862*t^4 - 2652*t^3 + 14397*t^2 - 9922*t + 18553

Elkies primitive

$g(t)$:

t^15 + 3*t^14 + 51*t^13 + 67*t^12 + 969*t^11 - 33*t^10 + 10963*t^9 - 9729*t^8 + 96507*t^7 - 108631*t^6 + 580785*t^5 - 700503*t^4 + 2102099*t^3 - 1877667*t^2 + 3904161*t - 1164691

Elkies primitive

$h(t)$:

4591650240*t^6 + 5509980288*t^5 + 101934635328*t^4 - 58773123072*t^3 + 730072388160*t^2 - 1151585880192*t + 5029693672896

Definition

A Davenport-Stothers polynomial triple of half-degree $M$ is a triple $f,g,h\in\mathbb Q[t]$ with $h=f^3-g^2\ne0$, $\deg f=2M$, $\deg g=3M$ and $\deg h=M+1$. Each row stores one representative of a listed equivalence class of such triples, and the part is one of its three polynomials.

Parameters

$M$

— half-degree (a positive integer with $\deg f=2M$)

class

— equivalence class (one of the equivalence classes of triples of the given half-degree)

Formulas

(1)

$h(t)=f(t)^3-g(t)^2$.

(2)

The Belyi map attached to the Hall dessin of the triple is $f(t)^3/h(t)=1+g(t)^2/h(t)$ [1].

Comments

(3)

Davenport's polynomial bound says that if $\deg f=2M$ and $f^3\ne g^2$, then $\deg(f^3-g^2)\geq M+1$ [1]. The rows attain the bound, so $\deg h=M+1$ is the extremal case, the polynomial analogue of Hall's conjecture.

(4)

Triples $(f,g,h)$ and $(f_1,g_1,h_1)$ are equivalent in the sense of [1] if $f_1(t)=w^2f(ut+v)$, $g_1(t)=w^3g(ut+v)$ and $h_1(t)=w^6h(ut+v)$ for some $u,w\in\mathbb C^*$ and $v\in\mathbb C$. The table stores, for each listed class, a representative from [1] or [3], with the variable renamed to $t$ where needed. An equivalent representative can have different coefficients until this substitution is made.

(5)

For a triple, the rational function $f(t)^3/h(t)=1+g(t)^2/h(t)$ is a Belyi map [1]. Its dessin has $2M$ vertices over $0$, each of valence $3$, $3M$ vertices over $1$, each of valence $2$, and $M+1$ faces. Montanus's Hall tree is obtained from this dessin by removing the markings and cutting off the faces; it is a plane tree with $M+1$ leaves and $M-1$ trivalent internal vertices.

(6)

Montanus gives one deformation class, hence one equivalence class of triples, for each $M\leq4$, four for $M=5$, and in general $D(M)$, the number of triangulations of a convex $(M+1)$-gon up to rotation [1]. At $M=5$, two of the four classes are exchanged by complex conjugation, so neither has a real representative [1]; the two classes stored here are the rational Birch and Elkies examples [3].

References

[1]

Hans Montanus, Halltripels en kindertekeningen, Nieuw Archief voor Wiskunde (5) 7 (2006), no. 3, 172-176. https://www.nieuwarchief.nl/serie5/pdf/naw5-2006-07-3-172.pdf

[2]

Jeroen Sijsling and John Voight, On computing Belyi maps, Publications mathematiques de Besancon (2014), 73-131. (arXiv)

[3]

Noam D. Elkies, Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, Algorithmic Number Theory (ANTS-IV), Lecture Notes in Computer Science 1838 (2000), 33-63. (arXiv)

Links

[4]

Wikipedia: Hall's conjecture

[5]

Wikipedia: Mason-Stothers theorem

Similar tables

Good examples of Hall's conjecture — records integer near-collisions of squares and cubes rather than polynomial ones

Known solutions of the Fermat-Catalan equation — records coprime perfect powers whose sum is a perfect power exactly

Counterexamples to Euler's sum of powers conjecture — records integer solutions of $\sum a_i^k=b^k$ with fewer than $k$ terms

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds the rational classes listed in [3]: the unique class for each $M\leq4$ and the Birch and Elkies classes at $M=5$; Elkies states that the existence of any rational class with $M>5$ is open)

Sources of data: [1], [2], [3]

How they were obtained:

Every stored value is an exact polynomial over $\mathbb Q$.

The generator transcribes $f$ from [1] for the Montanus rows and from [3] for the Elkies row, derives $g$ as the polynomial part of $f^{3/2}$ at infinity, sets $h=f^3-g^2$, and checks the degree conditions and the extremality criterion $2fg'-3f'g\in\mathbb Q^*$ exactly. The Birch $M=5$ row is also checked against Example 1.8 of [2], which prints $f$, $g$ and $h$.