Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds for the greatest common
divisor of the degrees of possible
counterexamples to the Jacobian Conjecture
Lower bounds
Juan José Guccione
(Joint work with J.A. Guccione and C. Valqui)
Departamento de Matemática
FCEN UBA
Lower bounds for
the greatest
common divisor
Outline
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminares
Lower bounds for
the greatest
common divisor
Outline
Preliminares
Shapes of
Jacobian Pairs
Preliminares
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Lower bounds for
the greatest
common divisor
Outline
Preliminares
Shapes of
Jacobian Pairs
Preliminares
Cutting the lower
edge
Standard minimal
pairs
Shapes of Jacobian Pairs
Lower bounds
Cutting the lower edge
Lower bounds for
the greatest
common divisor
Outline
Preliminares
Shapes of
Jacobian Pairs
Preliminares
Cutting the lower
edge
Standard minimal
pairs
Shapes of Jacobian Pairs
Lower bounds
Cutting the lower edge
Standard minimal pairs
Lower bounds for
the greatest
common divisor
Outline
Preliminares
Shapes of
Jacobian Pairs
Preliminares
Cutting the lower
edge
Standard minimal
pairs
Shapes of Jacobian Pairs
Lower bounds
Cutting the lower edge
Standard minimal pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
I
K an algebraically closed characteristic 0 field.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
I
L(l) := K [x l , x
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
1
−1
l
, y ].
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
I
L(l) := K [x l , x
I
V := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1}.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
1
−1
l
, y ].
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
I
L(l) := K [x l , x
I
V := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1}.
I
V≥0 := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1 and ρ + σ ≥ 0}.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
1
−1
l
, y ].
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
I
L(l) := K [x l , x
I
V := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1}.
I
V≥0 := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1 and ρ + σ ≥ 0}.
I
V>0 := {(ρ, σ) ∈ V≥0 : ρ + σ > 0}.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
1
−1
l
, y ].
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
I
K an algebraically closed characteristic 0 field.
I
L := K [x, y ] the polynomial K -algebra in two variables.
I
L(l) := K [x l , x
I
V := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1}.
I
V≥0 := {(ρ, σ) ∈ Z2 : gcd(ρ, σ) = 1 and ρ + σ ≥ 0}.
I
V>0 := {(ρ, σ) ∈ V≥0 : ρ + σ > 0}.
I
vρ,σ (i, j) := ρi + σj for each (i, j) ∈ 1l Z × N0 .
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
1
−1
l
, y ].
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
Let
(ρ, σ) ∈ V
and
P=
X
aij x i y j ∈ L(l) \ {0}.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let
Shapes of
Jacobian Pairs
(ρ, σ) ∈ V
and
P=
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
I
Supp(P) := {(i, j) : aij 6= 0}.
X
aij x i y j ∈ L(l) \ {0}.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let
Shapes of
Jacobian Pairs
(ρ, σ) ∈ V
and
P=
X
aij x i y j ∈ L(l) \ {0}.
Cutting the lower
edge
Standard minimal
pairs
I
Supp(P) := {(i, j) : aij 6= 0}.
I
vρ,σ (P) := max {vρ,σ (i, j) : aij 6= 0}.
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let
Shapes of
Jacobian Pairs
(ρ, σ) ∈ V
and
P=
X
aij x i y j ∈ L(l) \ {0}.
Cutting the lower
edge
Standard minimal
pairs
I
Supp(P) := {(i, j) : aij 6= 0}.
I
vρ,σ (P) := max {vρ,σ (i, j) : aij 6= 0}.
I
`ρ,σ (P) :=
Lower bounds
P
{ρi+σj=vρ,σ (P)} aij x
iyj.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let
Shapes of
Jacobian Pairs
(ρ, σ) ∈ V
and
P=
X
aij x i y j ∈ L(l) \ {0}.
Cutting the lower
edge
Standard minimal
pairs
I
Supp(P) := {(i, j) : aij 6= 0}.
I
vρ,σ (P) := max {vρ,σ (i, j) : aij 6= 0}.
I
`ρ,σ (P) :=
Lower bounds
P
{ρi+σj=vρ,σ (P)} aij x
iyj.
Definition 1
P ∈ L(l) is (ρ, σ)-homogeneous if P = `ρ,σ (P).
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
Let P ∈ L(l) \ {0}.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let P ∈ L(l) \ {0}.
Shapes of
Jacobian Pairs
Definition 2
Cutting the lower
edge
The Newton polygon of P is the convex hull of its support.
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let P ∈ L(l) \ {0}.
Shapes of
Jacobian Pairs
Definition 2
Cutting the lower
edge
The Newton polygon of P is the convex hull of its support.
Standard minimal
pairs
Lower bounds
Notation 3
Let (ρ, σ) ∈ V. We will denote by
stρ,σ (P)
and
enρ,σ (P)
the first and last point that one finds on Supp(`ρ,σ (P)), when
you scroll the Newton polygon of P in a counterclockwise way.
Lower bounds for
the greatest
common divisor
Preliminares
Preliminaries
Let P ∈ L(l) \ {0}.
Shapes of
Jacobian Pairs
Definition 2
Cutting the lower
edge
The Newton polygon of P is the convex hull of its support.
Standard minimal
pairs
Lower bounds
Notation 3
Let (ρ, σ) ∈ V. We will denote by
stρ,σ (P)
and
enρ,σ (P)
the first and last point that one finds on Supp(`ρ,σ (P)), when
you scroll the Newton polygon of P in a counterclockwise way.
Definition 4
The set Dir(P), of directions associated with P, is defined by
Dir(P) := {(ρ, σ) ∈ V : # Supp(`ρ,σ (P)) > 1}.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
P(x, y ) =2x −7 y 3 + 5x 12 y 8 − 3x −9 y 11 − x −2 y 15 − 6x 8 y 11
+ 2x 12 y 8 − x 8 y 4 − 3x −3 y 2 + 3xy 5 − 8x 3 y 13
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Preliminaries
P(x, y ) =2x −7 y 3 + 5x 12 y 8 − 3x −9 y 11 − x −2 y 15 − 6x 8 y 11
+ 2x 12 y 8 − x 8 y 4 − 3x −3 y 2 + 3xy 5 − 8x 3 y 13
Newton polygon of P
y
en2,5 (P)
st2,5 (P)
(2,5)
x
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Definition 5
Shapes of
Jacobian Pairs
V≥0 if ordered by
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
(ρ, σ) < (ρ0 , σ 0 )
if the argument of (ρ, σ) is less than that of (ρ0 , σ 0 ), where
the arguments are taken between −π/4 and 3π/4.
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Definition 5
Shapes of
Jacobian Pairs
V≥0 if ordered by
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
(ρ, σ) < (ρ0 , σ 0 )
if the argument of (ρ, σ) is less than that of (ρ0 , σ 0 ), where
the arguments are taken between −π/4 and 3π/4.
y
The directions grow
counterclockwise.
(−1,1)
x
(1,−1)
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Notation 6
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) , we write
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
[P, Q] := det J(P, Q),
where J(P, Q) is the jacobian matrix of (P, Q).
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Notation 6
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) , we write
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
[P, Q] := det J(P, Q),
where J(P, Q) is the jacobian matrix of (P, Q).
Definition 7
A, B ∈ R2 are aligned if both are in a straight line through
zero. In such a case we write A ∼ B.
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Notation 6
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) , we write
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
[P, Q] := det J(P, Q),
where J(P, Q) is the jacobian matrix of (P, Q).
Definition 7
A, B ∈ R2 are aligned if both are in a straight line through
zero. In such a case we write A ∼ B.
Remark 8
∼ is not an equivalence relation (it is so restricted to
R2 \ {0}).
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Proposition 9
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) \ {0} and (ρ, σ) ∈ V,
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
vρ,σ ([P, Q]) ≤ vρ,σ (P) + vρ,σ (Q) − (ρ + σ).
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Proposition 9
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) \ {0} and (ρ, σ) ∈ V,
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
vρ,σ ([P, Q]) ≤ vρ,σ (P) + vρ,σ (Q) − (ρ + σ).
Moreover the equality holds iff [`ρ,σ (P), `ρ,σ (Q)] 6= 0 and in
this case
[`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ ([P, Q]).
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Proposition 9
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) \ {0} and (ρ, σ) ∈ V,
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
vρ,σ ([P, Q]) ≤ vρ,σ (P) + vρ,σ (Q) − (ρ + σ).
Moreover the equality holds iff [`ρ,σ (P), `ρ,σ (Q)] 6= 0 and in
this case
[`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ ([P, Q]).
Example 10
If [P, Q] = 1, then vρ,σ (P) + vρ,σ (Q) ≥ (ρ + σ). Furthermore,
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Proposition 9
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) \ {0} and (ρ, σ) ∈ V,
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
vρ,σ ([P, Q]) ≤ vρ,σ (P) + vρ,σ (Q) − (ρ + σ).
Moreover the equality holds iff [`ρ,σ (P), `ρ,σ (Q)] 6= 0 and in
this case
[`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ ([P, Q]).
Example 10
If [P, Q] = 1, then vρ,σ (P) + vρ,σ (Q) ≥ (ρ + σ). Furthermore,
vρ,σ (P) + vρ,σ (Q) = ρ + σ ⇔ [`ρ,σ (P), `ρ,σ (Q)] = 1,
vρ,σ (P) + vρ,σ (Q) > ρ + σ ⇔ [`ρ,σ (P), `ρ,σ (Q)] = 0.
Lower bounds for
the greatest
common divisor
Preliminaries
Preliminares
Proposition 9
Shapes of
Jacobian Pairs
For P, Q ∈ L(l) \ {0} and (ρ, σ) ∈ V,
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
vρ,σ ([P, Q]) ≤ vρ,σ (P) + vρ,σ (Q) − (ρ + σ).
Moreover the equality holds iff [`ρ,σ (P), `ρ,σ (Q)] 6= 0 and in
this case
[`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ ([P, Q]).
Example 10
If [P, Q] = 1, then vρ,σ (P) + vρ,σ (Q) ≥ (ρ + σ). Furthermore,
vρ,σ (P) + vρ,σ (Q) = ρ + σ ⇔ [`ρ,σ (P), `ρ,σ (Q)] = 1,
vρ,σ (P) + vρ,σ (Q) > ρ + σ ⇔ [`ρ,σ (P), `ρ,σ (Q)] = 0.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
2. If [P, Q] = 0 and (µ, τ ) 6= (0, 0), then there exist
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
2. If [P, Q] = 0 and (µ, τ ) 6= (0, 0), then there exist
I
R ∈ L(l) ,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
2. If [P, Q] = 0 and (µ, τ ) 6= (0, 0), then there exist
I
R ∈ L(l) ,
I
λP , λQ ∈ K × ,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
2. If [P, Q] = 0 and (µ, τ ) 6= (0, 0), then there exist
I
R ∈ L(l) ,
I
λP , λQ ∈ K × ,
I
m, n ∈ Z coprime satisfying nτ = mµ,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(ρ, σ) ∈ V,
Shapes of
Jacobian Pairs
P, Q ∈ L(l) \ {0} two (ρ, σ)-homogeneous elements,
Cutting the lower
edge
τ := vρ,σ (P) and µ := vρ,σ (Q).
Standard minimal
pairs
Proposition 11
Lower bounds
1. If τ = µ = 0, then [P, Q] = 0.
2. If [P, Q] = 0 and (µ, τ ) 6= (0, 0), then there exist
I
R ∈ L(l) ,
I
λP , λQ ∈ K × ,
I
m, n ∈ Z coprime satisfying nτ = mµ,
such that
P = λP R m
and
Q = λQ R n
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
(1)
y
(ρ,σ)
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
(1)
y
Shapes of
Jacobian Pairs
Cutting the lower
edge
(ρ,σ)
Standard minimal
pairs
x
Lower bounds
(2)
y
Q
P
R
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Let P, Q, T ∈ L(l) \ {0} and (ρ, σ) ∈ V.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
Let P, Q, T ∈ L(l) \ {0} and (ρ, σ) ∈ V.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Proposition 12
If [`ρ,σ (P), `ρ,σ (Q)] = 0, then
stρ,σ (P) ∼ stρ,σ (Q)
and
enρ,σ (P) ∼ enρ,σ (Q).
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of Jacobian Pairs
Let P, Q, T ∈ L(l) \ {0} and (ρ, σ) ∈ V.
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Proposition 12
If [`ρ,σ (P), `ρ,σ (Q)] = 0, then
stρ,σ (P) ∼ stρ,σ (Q)
and
enρ,σ (P) ∼ enρ,σ (Q).
Proposition 13
If [`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ (T ), then
stρ,σ (P) stρ,σ (Q) iff stρ,σ (P)+stρ,σ (Q)−(1, 1) = stρ,σ (T ),
and similarly for enρ,σ .
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
[`ρ,σ (P), `ρ,σ (Q)] = 0.
Shapes of
Jacobian Pairs
Cutting the lower
edge
y
Standard minimal
pairs
enρ,σ (Q)
Lower bounds
`ρ,σ (Q)
enρ,σ (P)
`ρ,σ (P)
(ρ,σ)
stρ,σ (P)
stρ,σ (Q)
x
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
[`ρ,σ (P), `ρ,σ (Q)] = `ρ,σ (T ).
Shapes of
Jacobian Pairs
Cutting the lower
edge
y
Standard minimal
pairs
enρ,σ (T )
enρ,σ (P)+enρ,σ (Q)
Lower bounds
`ρ,σ (T )
`ρ,σ (Q)
enρ,σ (Q)
enρ,σ (P)
`ρ,σ (P)
stρ,σ (P)+stρ,σ (Q)
(ρ,σ)
stρ,σ (P)
stρ,σ (Q)
stρ,σ (T )
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Let P ∈ L(l) and let (ρ, σ) ∈ V>0 be such that vρ,σ (P) > 0.
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Let P ∈ L(l) and let (ρ, σ) ∈ V>0 be such that vρ,σ (P) > 0.
Shapes of
Jacobian Pairs
Theorem 14
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
If [P, Q] = 1 for some Q ∈ L(l) ,
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Let P ∈ L(l) and let (ρ, σ) ∈ V>0 be such that vρ,σ (P) > 0.
Shapes of
Jacobian Pairs
Theorem 14
Cutting the lower
edge
If [P, Q] = 1 for some Q ∈ L(l) , then there exist
Standard minimal
pairs
G0 ∈ K [P, Q] \ {0}
Lower bounds
such that
and F ∈ L(l)
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Let P ∈ L(l) and let (ρ, σ) ∈ V>0 be such that vρ,σ (P) > 0.
Shapes of
Jacobian Pairs
Theorem 14
Cutting the lower
edge
If [P, Q] = 1 for some Q ∈ L(l) , then there exist
Standard minimal
pairs
G0 ∈ K [P, Q] \ {0}
and F ∈ L(l)
Lower bounds
such that
F is (ρ, σ)-homogeneous,
vρ,σ (F ) = ρ + σ,
[F , `ρ,σ (P)] = `ρ,σ (P)
and
[`ρ,σ (G0 ), `ρ,σ (P)]F = `ρ,σ (P)`ρ,σ (G0 ).
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
1. If P, Q ∈ L, then we can take F ∈ L.
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
1. If P, Q ∈ L, then we can take F ∈ L.
Standard minimal
pairs
Lower bounds
2. stρ,σ (P) ∼ stρ,σ (F ) or stρ,σ (F ) = (1, 1).
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
1. If P, Q ∈ L, then we can take F ∈ L.
Standard minimal
pairs
Lower bounds
2. stρ,σ (P) ∼ stρ,σ (F ) or stρ,σ (F ) = (1, 1).
3. enρ,σ (P) ∼ enρ,σ (F ) or enρ,σ (F ) = (1, 1).
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
1. If P, Q ∈ L, then we can take F ∈ L.
Standard minimal
pairs
Lower bounds
2. stρ,σ (P) ∼ stρ,σ (F ) or stρ,σ (F ) = (1, 1).
3. enρ,σ (P) ∼ enρ,σ (F ) or enρ,σ (F ) = (1, 1).
4. stρ,σ (P) (1, 1) enρ,σ (P).
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Remark 15
Shapes of
Jacobian Pairs
In the previous theorem:
Cutting the lower
edge
1. If P, Q ∈ L, then we can take F ∈ L.
Standard minimal
pairs
Lower bounds
2. stρ,σ (P) ∼ stρ,σ (F ) or stρ,σ (F ) = (1, 1).
3. enρ,σ (P) ∼ enρ,σ (F ) or enρ,σ (F ) = (1, 1).
4. stρ,σ (P) (1, 1) enρ,σ (P).
5. If we define recursively Gi := [Gi−1 , P], then
[`ρ,σ (Gi ), `ρ,σ (P)] = 0 for i ≥ 1.
Lower bounds for
the greatest
common divisor
Shapes of Jacobian Pairs
Preliminares
Shapes of
Jacobian Pairs
y
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
enρ,σ (P)
`ρ,σ (P)
(ρ,σ)
F
stρ,σ (F )
enρ,σ (F )
stρ,σ (P)
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Let
(ρ, σ) ∈ V>0 with ρ > 0
and
P ∈ L(l) \ {0}.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Shapes of Jacobian Pairs
Let
(ρ, σ) ∈ V>0 with ρ > 0
and
P ∈ L(l) \ {0}.
Assume that vρ,σ (P) > 0 and that
F ∈ L(l) \ {0}
Lower bounds
is a vρ,σ -homogeneous element that satisfies
[F , `ρ,σ (P)] = `ρ,σ (P).
(1)
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Shapes of Jacobian Pairs
Let
(ρ, σ) ∈ V>0 with ρ > 0
and
P ∈ L(l) \ {0}.
Assume that vρ,σ (P) > 0 and that
Standard minimal
pairs
F ∈ L(l) \ {0}
Lower bounds
is a vρ,σ -homogeneous element that satisfies
[F , `ρ,σ (P)] = `ρ,σ (P).
Note that
vρ,σ (F ) = ρ + σ.
(1)
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Shapes of Jacobian Pairs
Let
(ρ, σ) ∈ V>0 with ρ > 0
and
P ∈ L(l) \ {0}.
Assume that vρ,σ (P) > 0 and that
Standard minimal
pairs
F ∈ L(l) \ {0}
Lower bounds
is a vρ,σ -homogeneous element that satisfies
[F , `ρ,σ (P)] = `ρ,σ (P).
(1)
Note that
vρ,σ (F ) = ρ + σ.
Write
r
u
`ρ,σ (P) = x l y s p(z) and F = x l y v f (z)
with z := x
− σρ
y and p(0) 6= 0 6= f (0).
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
1. f is separable and every irreducible factor of p divides f .
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
1. f is separable and every irreducible factor of p divides f .
2. If (ρ, σ) ∈ Dir(P), then v0,1 (stρ,σ (F )) < v0,1 (enρ,σ (F )).
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
1. f is separable and every irreducible factor of p divides f .
2. If (ρ, σ) ∈ Dir(P), then v0,1 (stρ,σ (F )) < v0,1 (enρ,σ (F )).
3. If p(z) = p(z k ) and f (z) = f (z k ), then f is separable
and every irreducible factor of p divides f .
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
1. f is separable and every irreducible factor of p divides f .
2. If (ρ, σ) ∈ Dir(P), then v0,1 (stρ,σ (F )) < v0,1 (enρ,σ (F )).
3. If p(z) = p(z k ) and f (z) = f (z k ), then f is separable
and every irreducible factor of p divides f .
4. If P, F ∈ L and v0,1 (enρ,σ (F )) − v0,1 (stρ,σ (F )) = ρ, then
the multiplicity of each linear factor of p is
1
1
deg(p) = v0,1 (enρ,σ (P)) − v0,1 (stρ,σ (P)) .
ρ
ρ
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Shapes of Jacobian Pairs
Proposition 16
1. f is separable and every irreducible factor of p divides f .
2. If (ρ, σ) ∈ Dir(P), then v0,1 (stρ,σ (F )) < v0,1 (enρ,σ (F )).
3. If p(z) = p(z k ) and f (z) = f (z k ), then f is separable
and every irreducible factor of p divides f .
4. If P, F ∈ L and v0,1 (enρ,σ (F )) − v0,1 (stρ,σ (F )) = ρ, then
the multiplicity of each linear factor of p is
1
1
deg(p) = v0,1 (enρ,σ (P)) − v0,1 (stρ,σ (P)) .
ρ
ρ
5. Assume (ρ, σ) ∈ Dir(P). If s > 0 or # factors(p) > 1,
then F satisfying (1) is unique.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Cutting the lower edge
For each λ ∈ K , k ∈ Z and l ∈ N, let ϕ ∈ Aut(L(l) ) be the
automorphism defined by
1
1
ϕ(x l ) := x l
and
k
ϕ(y ) := y + λx l .
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower edge
For each λ ∈ K , k ∈ Z and l ∈ N, let ϕ ∈ Aut(L(l) ) be the
automorphism defined by
1
Cutting the lower
edge
1
ϕ(x l ) := x l
and
k
ϕ(y ) := y + λx l .
Standard minimal
pairs
Lower bounds
Proposition 17
There exists a direction (ρ, σ), with ρ > 0 such that
σ
ρ
= kl ,
`ρ,σ (ϕ(P)) = ϕ(`ρ,σ (P)),
`−ρ,−σ (ϕ(P)) = ϕ(`−ρ,−σ (P))
and
`ρ1 ,σ1 (ϕ(P)) = `ρ1 ,σ1 (P).
for all P ∈ L(l) \ {0} and all (ρ, σ) < (ρ1 , σ1 ) < (−ρ, −σ).
Lower bounds for
the greatest
common divisor
Preliminares
Cutting the lower edge
Let λ, k, l, ϕ and (ρ, σ) be as before. Write
r
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
`ρ,σ (P) = x l y s p(z),
with z := x
− σρ
y and p(0) 6= 0. Let p(z) := z s p(z).
Lower bounds for
the greatest
common divisor
Preliminares
Cutting the lower edge
Let λ, k, l, ϕ and (ρ, σ) be as before. Write
r
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
`ρ,σ (P) = x l y s p(z),
with z := x
− σρ
y and p(0) 6= 0. Let p(z) := z s p(z).
Definition 18
ϕ is associated with P and (ρ, σ) if the multiplicity mλ of
z − λ in p(z) is maximum.
Lower bounds for
the greatest
common divisor
Preliminares
Cutting the lower edge
Let λ, k, l, ϕ and (ρ, σ) be as before. Write
r
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
`ρ,σ (P) = x l y s p(z),
with z := x
− σρ
y and p(0) 6= 0. Let p(z) := z s p(z).
Definition 18
ϕ is associated with P and (ρ, σ) if the multiplicity mλ of
z − λ in p(z) is maximum.
Now let P, Q ∈ L(l) . Assume that ϕ is associated with P and
(ρ, σ). Assume also that
vρ,σ (P)
m
=
> 0,
vρ,σ (Q)
n
with m, n ∈ N coprime.
Let mλ be the multiplicity of z − λ in p(z).
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
(a) [P, Q] = 1,
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
(d) vρ,σ (P) + vρ,σ (Q) > ρ + σ,
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
(d) vρ,σ (P) + vρ,σ (Q) > ρ + σ,
(e) v1,−1 enρ,σ (P) < 0.
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
(d) vρ,σ (P) + vρ,σ (Q) > ρ + σ,
(e) v1,−1 enρ,σ (P) < 0.
Then there exists (ρ0 , σ 0 ) ∈ V, with ρ0 > 0, such that
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
(d) vρ,σ (P) + vρ,σ (Q) > ρ + σ,
(e) v1,−1 enρ,σ (P) < 0.
Then there exists (ρ0 , σ 0 ) ∈ V, with ρ0 > 0, such that
1. Predϕ(P) (ρ, σ) = (ρ0 , σ 0 ).
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Proposition 19
Shapes of
Jacobian Pairs
Assume
Cutting the lower
edge
(a) [P, Q] = 1,
Standard minimal
pairs
(b) (ρ, σ) ∈ Dir(P) ∪ Dir(Q),
Lower bounds
(c) vρ,σ (P) > 0 and vρ,σ (Q) > 0,
(d) vρ,σ (P) + vρ,σ (Q) > ρ + σ,
(e) v1,−1 enρ,σ (P) < 0.
Then there exists (ρ0 , σ 0 ) ∈ V, with ρ0 > 0, such that
1. Predϕ(P) (ρ, σ) = (ρ0 , σ 0 ).
2. v1,−1 enρ0 ,σ0 (ϕ(P)) < 0 and v1,−1 enρ0 ,σ0 (ϕ(Q)) < 0.
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
3. vρ0 ,σ0 (ϕ(P)) > 0 and vρ0 ,σ0 (ϕ(Q)) > 0.
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
3. vρ0 ,σ0 (ϕ(P)) > 0 and vρ0 ,σ0 (ϕ(Q)) > 0.
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
4.
vρ0 ,σ0 (ϕ(P))
vρ0 ,σ0 (ϕ(Q))
=
vρ,σ (P)
vρ,σ (Q) .
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
3. vρ0 ,σ0 (ϕ(P)) > 0 and vρ0 ,σ0 (ϕ(Q)) > 0.
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
4.
vρ0 ,σ0 (ϕ(P))
vρ0 ,σ0 (ϕ(Q))
=
vρ,σ (P)
vρ,σ (Q) .
5. For all (ρ, σ) < (ρ00 , σ 00 ) < (−1, 1), the equalities
`ρ00 ,σ00 (ϕ(P)) = `ρ00 ,σ00 (P) and `ρ00 ,σ00 (ϕ(Q)) = `ρ00 ,σ00 (Q)
hold.
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
3. vρ0 ,σ0 (ϕ(P)) > 0 and vρ0 ,σ0 (ϕ(Q)) > 0.
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
4.
vρ0 ,σ0 (ϕ(P))
vρ0 ,σ0 (ϕ(Q))
=
vρ,σ (P)
vρ,σ (Q) .
5. For all (ρ, σ) < (ρ00 , σ 00 ) < (−1, 1), the equalities
`ρ00 ,σ00 (ϕ(P)) = `ρ00 ,σ00 (P) and `ρ00 ,σ00 (ϕ(Q)) = `ρ00 ,σ00 (Q)
hold.
6. enρ0 ,σ0 (ϕ(P)) = stρ,σ (ϕ(P)) =
enρ,σ (ϕ(P)) = enρ,σ (P).
r
l
+
sσ
ρ
− mλ σρ , mλ and
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
3. vρ0 ,σ0 (ϕ(P)) > 0 and vρ0 ,σ0 (ϕ(Q)) > 0.
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
4.
vρ0 ,σ0 (ϕ(P))
vρ0 ,σ0 (ϕ(Q))
=
vρ,σ (P)
vρ,σ (Q) .
5. For all (ρ, σ) < (ρ00 , σ 00 ) < (−1, 1), the equalities
`ρ00 ,σ00 (ϕ(P)) = `ρ00 ,σ00 (P) and `ρ00 ,σ00 (ϕ(Q)) = `ρ00 ,σ00 (Q)
hold.
6. enρ0 ,σ0 (ϕ(P)) = stρ,σ (ϕ(P)) =
enρ,σ (ϕ(P)) = enρ,σ (P).
7. enρ0 ,σ0 (ϕ(Q)) = stρ,σ (ϕ(Q))).
r
l
+
sσ
ρ
− mλ σρ , mλ and
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
[P, Q] = 1,
[`ρ,σ (P), `ρ,σ (Q)] = 0,
Cutting the lower
edge
y
Standard minimal
pairs
Lower bounds
`ρ,σ (Q)
`ρ,σ (P)
(ρ,σ)
x
Lower bounds for
the greatest
common divisor
Cutting the lower edge
Preliminares
Shapes of
Jacobian Pairs
[P, Q] = 1,
[`ρ,σ (P), `ρ,σ (Q)] = 0,
Cutting the lower
edge
y
Standard minimal
pairs
Lower bounds
`ρ,σ (ϕ(Q))
`ρ,σ (ϕ(P))
(ρ0 ,σ 0 )
(ρ,σ)
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Standard minimal pairs
(
∞
B :=
min gcd(v1,1 (P), v1,1 (Q))
where (P, Q) run on the counterexamples.
if JC is true,
if it is false,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal pairs
(
∞
B :=
min gcd(v1,1 (P), v1,1 (Q))
if JC is true,
if it is false,
where (P, Q) run on the counterexamples.
Standard minimal
pairs
Lower bounds
Definition 20
A minimal pair is a counterexample (P, Q) such that
B = gcd(v1,1 (P), v1,1 (Q)).
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal pairs
(
∞
B :=
min gcd(v1,1 (P), v1,1 (Q))
if JC is true,
if it is false,
where (P, Q) run on the counterexamples.
Standard minimal
pairs
Lower bounds
Definition 20
A minimal pair is a counterexample (P, Q) such that
B = gcd(v1,1 (P), v1,1 (Q)).
Proposition 21
If (P, Q) is a minimal pair,
v1,1 (P) 6 |v1,1 (Q)
and
v1,1 (Q) 6 |v1,1 (P).
Lower bounds for
the greatest
common divisor
Standard minimal pairs
Preliminares
Definition 22
Shapes of
Jacobian Pairs
Let m, n ∈ N be coprime with n, m > 1 and let P, Q ∈ L(l) .
We say that (P, Q) is an (m, n)-pair if
Cutting the lower
edge
Standard minimal
pairs
[P, Q] = 1,
Lower bounds
v1,0 (P)
v1,1 (P)
m
=
=
v1,1 (Q)
v1,0 (Q)
n
and
v1,−1 (en1,0 (P)) < 0.
Lower bounds for
the greatest
common divisor
Standard minimal pairs
Preliminares
Definition 22
Shapes of
Jacobian Pairs
Let m, n ∈ N be coprime with n, m > 1 and let P, Q ∈ L(l) .
We say that (P, Q) is an (m, n)-pair if
Cutting the lower
edge
Standard minimal
pairs
[P, Q] = 1,
Lower bounds
v1,0 (P)
v1,1 (P)
m
=
=
v1,1 (Q)
v1,0 (Q)
n
and
v1,−1 (en1,0 (P)) < 0.
An (m, n)-pair (P, Q) ∈ L(1) is standard if
v1,−1 (st1,0 (P)) < 0.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Proposition 24
Let P, Q ∈ L(1) such that (P, Q) is a standard (m, n)-pair.
There exists (ρ, σ) ∈ V≥0 , with σ < 0, such that:
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Proposition 24
Let P, Q ∈ L(1) such that (P, Q) is a standard (m, n)-pair.
There exists (ρ, σ) ∈ V≥0 , with σ < 0, such that:
1. vρ,σ (P) > 0 and vρ,σ (Q) > 0,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Proposition 24
Let P, Q ∈ L(1) such that (P, Q) is a standard (m, n)-pair.
There exists (ρ, σ) ∈ V≥0 , with σ < 0, such that:
1. vρ,σ (P) > 0 and vρ,σ (Q) > 0,
2.
vρ,σ (P)
vρ,σ (Q)
=
m
n,
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Proposition 24
Let P, Q ∈ L(1) such that (P, Q) is a standard (m, n)-pair.
There exists (ρ, σ) ∈ V≥0 , with σ < 0, such that:
1. vρ,σ (P) > 0 and vρ,σ (Q) > 0,
2.
vρ,σ (P)
vρ,σ (Q)
=
m
n,
3. (ρ, σ) ∈ Dir(P) ∩ Dir(Q),
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Proposition 23
If B < ∞, then there is a minimal pair (P, Q) in L, that is a
standard (m, n)-pair.
Proposition 24
Let P, Q ∈ L(1) such that (P, Q) is a standard (m, n)-pair.
There exists (ρ, σ) ∈ V≥0 , with σ < 0, such that:
1. vρ,σ (P) > 0 and vρ,σ (Q) > 0,
2.
vρ,σ (P)
vρ,σ (Q)
=
m
n,
3. (ρ, σ) ∈ Dir(P) ∩ Dir(Q),
4. v1,−1 stρ,σ (P) > 0 and v1,−1 enρ,σ (P) < 0.
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
y
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
`ρ,σ (P)
`ρ,σ (Q)
x
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Let
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let
Shapes of
Jacobian Pairs
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Cutting the lower
edge
I
(ρ, σ) as in Proposition 24,
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let
Shapes of
Jacobian Pairs
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Cutting the lower
edge
I
(ρ, σ) as in Proposition 24,
Standard minimal
pairs
I
`ρ,σ (P) = x r y s p(z),
Lower bounds
with z = x −σ/ρ y and p(0) 6= 0,
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let
Shapes of
Jacobian Pairs
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Cutting the lower
edge
I
(ρ, σ) as in Proposition 24,
Standard minimal
pairs
I
`ρ,σ (P) = x r y s p(z),
I
mλ the highest multiplicity of a λ ∈ K in p := z s p(z).
Lower bounds
with z = x −σ/ρ y and p(0) 6= 0,
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let
Shapes of
Jacobian Pairs
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Cutting the lower
edge
I
(ρ, σ) as in Proposition 24,
Standard minimal
pairs
I
`ρ,σ (P) = x r y s p(z),
I
mλ the highest multiplicity of a λ ∈ K in p := z s p(z).
Lower bounds
with z = x −σ/ρ y and p(0) 6= 0,
Let ϕ and (ρ0 , σ 0 ) be as in Proposition 19 and set
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let
Shapes of
Jacobian Pairs
I
(P, Q) ∈ L(1) be a standard (m, n)-pair,
Cutting the lower
edge
I
(ρ, σ) as in Proposition 24,
Standard minimal
pairs
I
`ρ,σ (P) = x r y s p(z),
I
mλ the highest multiplicity of a λ ∈ K in p := z s p(z).
Lower bounds
with z = x −σ/ρ y and p(0) 6= 0,
Let ϕ and (ρ0 , σ 0 ) be as in Proposition 19 and set
1
enρ,σ (P),
m
1
A00 := stρ,σ (P),
m
1
A1 := enρ0 ,σ0 (ϕ(P)).
m
A0 :=
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let F ∈ L be a (ρ, σ)-homogeneous polynomial such that
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
[P, F ]ρ,σ = `ρ,σ (P)
and
vρ,σ (F ) = ρ + σ.
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
Let F ∈ L be a (ρ, σ)-homogeneous polynomial such that
Shapes of
Jacobian Pairs
Cutting the lower
edge
[P, F ]ρ,σ = `ρ,σ (P)
and
vρ,σ (F ) = ρ + σ.
Standard minimal
pairs
Lower bounds
Write
(f1 , f2 ) := enρ,σ (F ),
(u, v ) := A0 ,
(r 0 , s 0 ) := A00
and
γ :=
mλ
.
m
Lower bounds for
the greatest
common divisor
Preliminares
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds
Theorem 25
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
1. s 0 < r 0 < u < v ,
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
5. uf2 = vf1 and ρ ≤ u.
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
5. uf2 = vf1 and ρ ≤ u.
1
6. (ρ, σ) = f2d−1 , 1−f
, where d := gcd(f1 − 1, f2 − 1).
d
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
5. uf2 = vf1 and ρ ≤ u.
1
6. (ρ, σ) = f2d−1 , 1−f
, where d := gcd(f1 − 1, f2 − 1).
d
7. A1 = A00 + (γ − s 0 ) − σρ , 1 .
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
5. uf2 = vf1 and ρ ≤ u.
1
6. (ρ, σ) = f2d−1 , 1−f
, where d := gcd(f1 − 1, f2 − 1).
d
7. A1 = A00 + (γ − s 0 ) − σρ , 1 .
8. A1 is not of the form t − ρ1 , t for any t ∈ N.
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Theorem 25
Shapes of
Jacobian Pairs
A0 , A00 ∈ N0 × N0 and vρ,σ (A0 ) = vρ,σ (A00 ). Moreover:
Cutting the lower
edge
1. s 0 < r 0 < u < v ,
Standard minimal
pairs
2. 2 ≤ f1 < u,
Lower bounds
3. gcd(u, v ) > 1,
4. enρ,σ (F ) = µA0
for some 0 < µ < 1,
5. uf2 = vf1 and ρ ≤ u.
1
6. (ρ, σ) = f2d−1 , 1−f
, where d := gcd(f1 − 1, f2 − 1).
d
7. A1 = A00 + (γ − s 0 ) − σρ , 1 .
8. A1 is not of the form t − ρ1 , t for any t ∈ N.
9. γ ≤ (v − s 0 )/ρ. If d = 1, then γ = (v − s 0 )/ρ.
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Shapes of
Jacobian Pairs
y
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
mv
mA0
mA1
mA00
ms 0
f2
f1
mr 0 mu
x
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Proposition 26
Shapes of
Jacobian Pairs
If A0 is as before Proposition 25, then v1,1 (A0 ) ≥ 16.
Cutting the lower
edge
Standard minimal
pairs
A0
(f1 , f2 )
(ρ, σ)
A00
d
γ
A1
(3,6)
(3,9)
(3,12)
(4,6)
(4,8)
(4,8)
(4,10)
(5,10)
(2,4)
(2,6)
(2,8)
(2,3)
(2,4)
(3,6)
(2,5)
(2,4)
(3,-1)
(5,-1)
(7,-1)
(2,-1)
(3,-1)
(5,-2)
(4,-1)
(3,-1)
(1,0)
×
×
(1,0)
×
×
×
(2,1)
1
2
2 − 31 , 2
1
3
3 − 12 , 3
1
3
3 − 31 , 3
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
A0
(f1 , f2 )
(ρ, σ)
A00
d
γ
A1
(5,10)
(5,10)
(6,8)
(6,9)
(6,9)
(3,6)
(4,8)
(3,4)
(2,3)
(4,6)
(5,-2)
(7,-3)
(3,-2)
(2,-1)
(5,-3)
(1,0)
×
×
(2,1)
×
1
2
2 − 51 , 2
1
4
4 − 12 , 4
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
A0
(f1 , f2 )
(ρ, σ)
A00
d
γ
A1
(5,10)
(5,10)
(6,8)
(6,9)
(6,9)
(3,6)
(4,8)
(3,4)
(2,3)
(4,6)
(5,-2)
(7,-3)
(3,-2)
(2,-1)
(5,-3)
(1,0)
×
×
(2,1)
×
1
2
2 − 51 , 2
1
4
4 − 12 , 4
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Corollary 27
B ≥ 16.
Lower bounds for
the greatest
common divisor
Preliminares
Lower bounds
A0
(f1 , f2 )
(ρ, σ)
A00
d
γ
A1
(5,10)
(5,10)
(6,8)
(6,9)
(6,9)
(3,6)
(4,8)
(3,4)
(2,3)
(4,6)
(5,-2)
(7,-3)
(3,-2)
(2,-1)
(5,-3)
(1,0)
×
×
(2,1)
×
1
2
2 − 51 , 2
1
4
4 − 12 , 4
Shapes of
Jacobian Pairs
Cutting the lower
edge
Standard minimal
pairs
Lower bounds
Corollary 27
B ≥ 16.
Proof
If (P, Q) is a standard minimal pair, then
1
1
B = v1,1 (P) ≥ v1,1 (enρ,σ (P)) = v1,1 (A0 ) ≥ 16.
m
m
Lower bounds for
the greatest
common divisor
Lower bounds
Preliminares
Proposition 28
Shapes of
Jacobian Pairs
Let (P, Q), with P, Q ∈ L(1) be a standard (m, n)-pair and let
A0 = (u, v ) be as above. Then
Cutting the lower
edge
Standard minimal
pairs
v ≤ u(u − 1),
u ≥ 4 and
Lower bounds
Remark 29
Write
enρ,σ (F ) =
Then q 6= 2.
p
A0 .
q
gcd(u, v ) > 2.