Post on 22-Aug-2020
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