Pairs of Polynomials Over theRationals Taking Infinitely Many

Common Values

Benjamin L. Weiss

Technion–Israel Institute of Technology

January 19th,2012.

Question. For which pairs of polynomials G(T ), H(T ) ∈ Q[T ]

do the multi-sets {G(x) | x ∈ Q} and {H(x) | x ∈ Q} haveinfinite intersection?

Equivalently:Question. For which pairs of polynomials G(T ), H(T ) ∈ Q[T ]

does G(X) = H(Y ) have infinitely many rational solutions?

m + pm + p + 1

m + p + 2 m + p - 1

2p

2p + 1 = n 2m

2m + 1

2k - 2

2

k

1

2k - 1

2 !

2m - 1

3

2k - 3 k + 1

k - 1

2 ! - 1

! + k - 1

2k

! + k

2 ! + 1

2m - 2

m + ! - 1

m + !

2 ! + 2

2m - 3

2 ! - 22k + 1

! + k - 2 ! + k + 1

Example. The equation X2 = 1−Y 2 has infinitely many ratio-

nal solutions parametrized by X =1− r2

1+ r2, Y =

2r

1+ r2.

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Theorem (Faltings’ Theorem 1983). If G(T ), H(T ) ∈ Q[T ] aresuch that G(X) − H(Y ) is absolutely irreducible and has in-finitely many zeros (x, y) ∈ Q2, then the curve defined byG(X) = H(Y ) has genus 0 or 1.

Example.

Xa(X − 1)m−a − cY a(Y − 1)m−a

with (m,a) = 1 and c 6= 0,1 is irreducible and defines a genus0 curve. The rational points on the curve are parametrized by

X =cuta − 1

cu+vtm − 1, Y = cvtm−aX with u, v ∈ Z such that

(m− a)u− av = 1.

Definition. Two pairs of polynomials (G,H) and (G, H) are(linearly) equivalent if there are linear polynomials µ, ψ, φ suchthat

G = µ ◦G ◦ ψ, H = µ ◦H ◦ φ,

or

G = µ ◦H ◦ ψ, H = µ ◦G ◦ φ.

Theorem (W. 2011). Let G(T ), H(T ) ∈ C[T ] such thatmax(deg(H),deg(G)) ≥ 11. Then G(X) − H(Y ) is abso-lutely irreducible and defines a genus zero curve if and only if(G(X), H(Y )) is (linearly) equivalent to a member of one ofsix infinite families. In all the infinite families, the smaller degreepolynomial is equivalent to one of

• Xm for any positive integer m;

• Xa (X − 1)m−a for any coprime integers a and m with1 ≤ a < m;

• Tm(X) where Tm is the degree m Chebyshev polynomial.

This extends previous results of Ritt, Bilu-Tichy, Beukers-Shorey-Tijdeman, Davenport-Lewis-Schinzel, and many others.

The six infinite families:

• Xm = (Y − y0)aF (Y )m with (m,a) = 1;

• Xm = (Y − y1)a(Y − y2)m−aF (Y )m with (m,a) = 1;

• Xa(X − 1)m−a = cY a(Y − 1)m−a with (m,a) = 1 andc 6= 0,1;

• Tm(X) = Tn(Y ) with (m,n) = 1;

• T2m(X) = −T2n(Y ) with (m,n) = 1;

• Tm(X) = H(Y ) with m|deg(H) and H(Y ) satisfies thefollowing polynomial Pell equation:

H(Y )2 − P (Y )Q(Y )2 = 4

with P (Y ) squarefree, deg(P ) = 4, and P (Y )Q(Y ) hav-ing at most 1 repeated root.

Polynomial Pell Equation

There are three subcases of the polynomial Pell equation:

• Q(Y ) has a repeated root;

• P (Y ) and Q(Y ) have a shared root;

• P (Y )Q(Y ) is squarefree.

Theorem. For any degree n there are n−624 J2(n)+

12φ(n) equiv-

alence classes of indecomposable polynomials in the first case,and 1

6J2(n) −12φ(n) equivalence classes in the second case.

We denote

Jk(n) = nk∏p|n

(1−1

pk)

The steps to prove the main theorem:• Riemann-Hurwitz equation expresses the genus of the curveG(X) = H(Y ) in terms of the ramification of the projectiononto the X (or Y ) projective line.• Since G(X) = H(Y ) has split variables, the ramification

of the projection is determined by the ramification inG,H : P1 → P1.• Combinatorially determine all ramification types correspond-

ing to G(X) = H(Y ) of genus 0 or 1.• Use topology of the punctured Riemann sphere to count the

number of polynomials with prescribed ramification.• Determine all decompositions of the polynomials in the infi-

nite families. Use the decompositions and a result of Friedto determine whether G(X)−H(Y ) is irreducible.

Riemann-HurwitzDefinition. Given a polynomial G(T ) ∈ C[T ], constant λ ∈ C,and factorization G(T )− λ =

∏ji=1(T − ci)

ei, the ramificationtype of G(T ) over λ is the multiset [e1, . . . , ei].

Definition. Given polynomials G(T ), H(T ) ∈ C[T ] the ramifi-cation type of G(T ) is the collection ramification types of G(T )

over all constants λ ∈ C. The multisetsA1, . . . Ak andB1, . . . , Bkwill be all the non-trivial ramification types of G(T ) and H(T )

with Ai and Bi the ramification over the same point.

Remark. Except for finitely many λ ∈ C, the ramification type ofG(T ) at λ is [1m] := [1,1, . . . ,1] (m times).

Definition. Given a rational function, we view it as a map fromP1 → P1 and define the ramification types of maps betweencurves similarly.

Theorem (Riemann-Hurwitz). Given two projective curvesC andD of respective genera gC, gD and F : C → D a rational mapof degree n with ramification structure A1, . . . , Ak,, it holds that

2gC − 2 = 2ngD − 2n+k∑i=1

∑a∈Ai

(a− 1).

Ramification of Projection

R1 . . . Rgcd(a,b)a

gcd(a,b)b

gcd(a,b) C(X,Y )φ

P

a

C(X)

G

C(Y )

H

Q

b

p C(T = G(X) = H(Y )) p

The ramification of φ is determined by the ramification of G andH.

CombinatoricsLemma. Given two polynomialsG(T ), H(T ) ∈ C[T ] of respec-tive degreesm and n such thatG(X)−H(Y ) is absolutely irre-ducible and defines a genus g curve, then the following relationshold:

1.∑a∈Ai

a = m, and∑b∈Bi

b = n for any 1 ≤ i ≤ k,

2.k∑i=1

∑a∈Ai

(a− 1) = m− 1, andk∑i=1

∑b∈Bi

(b− 1) = n− 1,

3. If G(T ) is not linearly equivalent to a perfect power, thengcd(a : a ∈ Ai) = 1 for all 1 ≤ i ≤ k.

4.k∑i=1

∑a∈Ai

∑b∈Bi

[a− gcd(a, b)] = m− 2+2g+gcd(m,n).

Bounding The Number of Branch Points

Definition. A constant λ ∈ C is a branch point (or critical value)of a polynomial G(T ) ∈ C[T ] if G(T )− λ has a repeated root.

Theorem (W.). IfG(T ), H(T ) ∈ C[T ] are such thatG(T ) is notequivalent to a perfect power and G(X) −H(Y ) is irreducibleand defines a genus 0 curve, thenH(T ) cannot have more than3 finite branch points. If in addition deg(H) ≥ 7 and H(T ) has3 finite branch points, then the polynomials’ ramification typesare:

A1 = A2 =[2m−1/2,1

], B1 = B2 =

[2n/2−1,12

],

B3 =[2,1n−2

].

Examples of Ramification Types for Two Branch Points

The following are examples of the infinite families of ramificationtypes whenG(X) andH(Y ) have exactly the same two branchpoints.

A1 = A2 =[2n−12 ,1

]and

• B1 =[4,2

m−52 ,1

], B2 =

[2m−32 ,13

];

• B1 =[4,2

m−72 ,13

], B2 =

[2m−12 ,1

];

• B1 =[3,2

m−32

], B2 =

[2m−32 ,13

];

• B1 =[3,2

m−52 ,12

], B2 =

[2m−12 ,1

];

Algebraic Topology At this point we have a list of (pairs of)ramification structures, and we need to determine the corre-sponding polynomials G,H, and to decide whether G(X)-H(Y) isirreducible. We use the Riemann Existence Theorem to countpolynomials with a given ramification type.

Theorem (Riemann Existence). Every “legal" ramification typeoccurs for at least one polynomial. Additionally, the numberof equivalence classes of polynomials with a given ramificationtype is related to the group of deck transformations of a cover ofthe punctured Riemann sphere.

Example. There are (n−5)(n−3)(n−1)48 equivalence classes of

polynomials with ramification type

B1 =[4,2

n−52 ,1

], B2 =

[2n−32 ,13

].

The theorem states that this is equivalent to counting (up toequivalency) the number of pairs (a, b) ∈ (Sn)2 with ab an n-cycle, a the product of a 4 cycle and n−5

2 2-cycles, and b theproduct of n−32 2-cycles. Two pairs (a, b) and (a, b) are equiva-lent if there is an element g ∈ Sn with

a = gag−1

and

b = gbg−1.

m + pm + p + 1

m + p + 2 m + p - 1

2p

2p + 1 = n 2m

2m + 1

2k - 2

2

k

1

2k - 1

2 !

2m - 1

3

2k - 3 k + 1

k - 1

2 ! - 1

! + k - 1

2k

! + k

2 ! + 1

2m - 2

m + ! - 1

m + !

2 ! + 2

2m - 3

2 ! - 22k + 1

! + k - 2 ! + k + 1

Theorem (Fried 1973). Given G(T ), H(T ) ∈ C[T ] there aredecompositions G = G1 ◦G2 and H = H1 ◦H2 such that thefactors of G(X)−H(Y ) are in one to one correspondence withthe factors of G1(X)−H1(Y ).

Corollary. If G(X)−H(Y ) is reducible, then there are decom-positions G = G1 ◦G2 and H = H1 ◦H2 such that• G1 and H1 have the same branch points;• The LCM of the ramification indices over any λ ∈ C of G1

and H1 are the same;• deg(G1) = deg(H1).

Decompositions of the Polynomials

We are able to show that every polynomial solution to a Pellequation decomposes as H(X) = ±Tk ◦ H(Y ) for some posi-tive odd integer k, and indecomposable polynomial H(Y ) whichis a solution to a corresponding Pell equation. Using this, wecan conclude (from Fried’s Theorem) that the curve defined byTm(X) = H(Y ) is irreducible when (m, k) = 1.Example. IfH(Y ) has branch points±2 and ramification struc-ture

B1 =[3,2

m−32

], B2 =

[2m−32 ,13

],

then H(Y ) decomposes as H(Y ) = ±Tk ◦ H(Y ) where H isindecomposable and has ramification structure

[3,2,2, . . . ,2] , [2,2, . . . ,2,1,1,1].

Polynomial Pell Equations with Solutions over Q

Theorem (Avanzi-Zannier 2001). The polynomials P (Y ) in theequation

H(Y )2 − P (Y )Q(Y )2 = 4

together with the indecomposable solutions toH(Y ) correspondto the isomorphism classes of elliptic curves with a torsion pointof order equal to the degree of H(Y ).

Theorem (Mazur 1977). No elliptic curve defined over Q has arational torsion point of order larger than 12.

To recap the infinite families again:

• Xm = (Y − y0)aF (Y )m with (m,a) = 1;

• Xm = (Y − y1)a(Y − y2)m−aF (Y )m with (m,a) = 1;

• Xa(X − 1)m−a = cY a(Y − 1)m−a with (m,a) = 1 andc 6= 0,1;

• Tm(X) = Tn(Y ) with (m,n) = 1;

• T2m(X) = −T2n(Y ) with (m,n) = 1;

• Tm(X) = H(Y ) with m|deg(H) and H(Y ) satisfies thefollowing polynomial Pell equation:

H(Y )2 − P (Y )Q(Y )2 = 4

with P (Y ) squarefree, deg(P ) = 4, and P (Y )Q(Y ) hav-ing at most 1 repeated root.

Future Work• We plan on extending the combinatorics to handle the case

where G(X) = H(Y ) defines a genus 1 curve.• There’s a method to extend to the case where G(X) =

H(Y ) is reducible.

C(X,Y )

C(X,H1(Y )) C(G1(X), Y )

C(X) C(G1(X), H1(Y )) C(Y )

C(G1(X)) C(H1(Y ))

C(t = G(X) = H(Y ))

Related Work• Lisa Berger studied the case with G(T ), H(T ) ∈ C(T )

such that G(T ) and H(T ) have only 0 and ∞ as com-mon branch points, and G(X) = H(Y ) defines a genus 1

curve.• Unique range sets of families of univariate functions from

C → C are sets for which the inverse image of the set un-der one of these functions uniquely determines the function.Understanding the rational components of curves G(X) =

cG(Y ) with genus 0 would help complete a classification ofunique range sets for polynomials.• Similar techniques have been used by A. Carney, R. Hortsch,

and M. Zieve to show that for any polynomial G(T ) ∈ Q[T ]

there are at most finitely many rational numbers r with∣∣∣G−1(r)⋂Q∣∣∣ > 6.

Thank You