Amoebas, ArchTrop and all that jazz
Roman Kogan
March 24, 2014
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
1 / 18
Overview of the talk
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Overview of the talk
Motivation and defintions
Newton Polyhedron
Amoeba
ArchTrop
Geometric results
Complexity results
Gnarly details
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
2 / 18
Motivation
Motivation: say something about zeros without evaluating
FTA: # zeros ≤ degree
Descartes: # real roots ≤ # coefficient sign changes
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
3 / 18
Motivation
Motivation: say something about zeros without evaluating
FTA: # zeros ≤ degree
Descartes: # real roots ≤ # coefficient sign changes
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
3 / 18
Motivation
Motivation: say something about zeros without evaluating
FTA: # zeros ≤ degree
Descartes: # real roots ≤ # coefficient sign changes
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
3 / 18
Newton Polyhedron
Newton polygon = generalization of the degree
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
4 / 18
Newton Polyhedron
Newton polygon = generalization of the degree
ai ∈ Zn
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
4 / 18
Newton Polyhedron
Newton polygon = generalization of the degree
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
=
τ
X
ai
a
ci x1 1 . . . xnin
i=1
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
4 / 18
Newton Polyhedron
Newton polygon = generalization of the degree
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
=
τ
X
ai
a
ci x1 1 . . . xnin
i=1
Newt(p) = Conv{a1 , a2 , . . . , aτ }
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
4 / 18
Newton Polyhedron
Newton polygon example
p(x) = x2 y + y 2 + y 4 + x4 y 4 + x3 y 3 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
5 / 18
Newton Polyhedron
Newton polygon example
p(x) = x2 y + y 2 + y 4 + x4 y 4 + x3 y 3 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
5 / 18
Newton Polyhedron
Newton polygon example
p(x) = x2 y + y 2 + y 4 + x4 y 4 + x3 y 3 ;
Newt(p) = Conv{(2, 1), (0, 2), (0, 4), (4, 4), (3, 3)}
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
5 / 18
Newton Polyhedron
Newton polygon example
p(x) = x2 y + y 2 + y 4 + x4 y 4 + x3 y 3 ;
Newt(p) = Conv{(2, 1), (0, 2), (0, 4), (4, 4), (3, 3)}
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
5 / 18
Amoeba
Q: What does the Newton polyhedron tell about the geometry of roots?
Amoeba: zero set on log paper
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
6 / 18
Amoeba
Q: What does the Newton polyhedron tell about the geometry of roots?
Amoeba: zero set on log paper
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
6 / 18
Amoeba
Q: What does the Newton polyhedron tell about the geometry of roots?
Amoeba: zero set on log paper
Z(p) := {(z1 , . . . , zn ) ∈ (C − 0)n : p(z1 , . . . , zn ) = 0}
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
6 / 18
Amoeba
Q: What does the Newton polyhedron tell about the geometry of roots?
Amoeba: zero set on log paper
Z(p) := {(z1 , . . . , zn ) ∈ (C − 0)n : p(z1 , . . . , zn ) = 0}
Amoeba(p) := {(log |z1 |, . . . , log |zn |) : (z1 , . . . , zn ) ∈ Z(p)}
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
6 / 18
Amoeba
Amoeba(p) := {(log |z1 |, . . . , log |zn |) : (z1 , . . . , zn ) ∈ Z(p)}
A simple example:
p(x) = x + y + 1;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
7 / 18
Amoeba
Amoeba(p) := {(log |z1 |, . . . , log |zn |) : (z1 , . . . , zn ) ∈ Z(p)}
A simple example:
p(x) = x + y + 1;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
7 / 18
Amoeba
Amoeba(p) := {(log |z1 |, . . . , log |zn |) : (z1 , . . . , zn ) ∈ Z(p)}
A simple example:
p(x) = x + y + 1;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
7 / 18
Amoeba
Amoeba(p) := {(log |z1 |, . . . , log |zn |) : (z1 , . . . , zn ) ∈ Z(p)}
A simple example:
p(x) = x + y + 1;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
7 / 18
Amoeba
Amoeba: more complicated example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
8 / 18
Amoeba
Amoeba: more complicated example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
8 / 18
Amoeba
Amoeba: more complicated example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
8 / 18
Amoeba
Amoeba: more complicated example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
8 / 18
Amoeba
Amoeba: more complicated example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4 ;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
8 / 18
ArchTrop
The missing link:
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop
The missing link:
ai ∈ Zn
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop
The missing link:
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop
The missing link:
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop
The missing link:
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchTrop := {(log |x1 |, . . . , log |xn |) ∈ Rn :
max |ci xai | is attained at least twice}
i
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop
The missing link:
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchTrop := {(log |x1 |, . . . , log |xn |) ∈ Rn :
max |ci xai | is attained at least twice}
i
log x ∈ ArchTrop ⇔ norms of at least two monomials of p(x) are equal
and larger than norms of other monomials at x
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
9 / 18
ArchTrop: simple example
ArchTrop := {log x ∈ Rn : max |xai | is attained at least twice}
i
Let p(x) = x + y + 1. Then ArchTrop = union of solutions to
(
(
(
log |x| = log |y|
log |x| = log |1|
log |y| = log |y|
log |x| ≥ log |1|
log |1| ≥ log |y|
log |1| ≥ log |x|
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
10 / 18
ArchTrop: simple example
ArchTrop := {log x ∈ Rn : max |xai | is attained at least twice}
i
Let p(x) = x + y + 1. Then ArchTrop = union of solutions to
(
(
(
log |x| = log |y|
log |x| = log |1|
log |y| = log |y|
log |x| ≥ log |1|
log |1| ≥ log |y|
log |1| ≥ log |x|
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
10 / 18
ArchTrop: simple example
ArchTrop := {log x ∈ Rn : max |xai | is attained at least twice}
i
Let p(x) = x + y + 1. Then ArchTrop = union of solutions to
(
(
(
log |x| = log |y|
log |x| = log |1|
log |y| = log |y|
log |x| ≥ log |1|
log |1| ≥ log |y|
log |1| ≥ log |x|
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
10 / 18
ArchTrop: simple example
ArchTrop := {log x ∈ Rn : max |xai | is attained at least twice}
i
Let p(x) = x + y + 1. Then ArchTrop = union of solutions to
(
(
(
log |x| = log |y|
log |x| = log |1|
log |y| = log |y|
log |x| ≥ log |1|
log |1| ≥ log |y|
log |1| ≥ log |x|
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
10 / 18
ArchTrop Example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
11 / 18
ArchTrop Example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
11 / 18
ArchTrop Example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
11 / 18
ArchTrop Example
p(x) = x2 y + y 2 + 3x2 y 3 + y 4 + x4 y 4
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
11 / 18
ArchNewt and ArchTrop
The missing link: duality
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchNewt(p) := Conv{(ai , − log |ci |) : i = 1..τ } ⊂ Rn+1
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchNewt(p) := Conv{(ai , − log |ci |) : i = 1..τ } ⊂ Rn+1
ArchTrop := {v ∈ Rn : (v, −1)
is maximal on a +dim lower face of ArchNewt}
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchNewt(p) := Conv{(ai , − log |ci |) : i = 1..τ } ⊂ Rn+1
ArchTrop := {v ∈ Rn : (v, −1)
is maximal on a +dim lower face of ArchNewt}
(v is an outer normal to a 1+-dimensional cell )
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
ArchNewt and ArchTrop
The missing link: duality
ai ∈ Zn
τ
X
p(x) =
ci xai
i=1
ArchNewt(p) := Conv{(ai , − log |ci |) : i = 1..τ } ⊂ Rn+1
ArchTrop := {v ∈ Rn : (v, −1)
is maximal on a +dim lower face of ArchNewt}
(v is an outer normal to a 1+-dimensional cell )
Easy to compute!
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
12 / 18
Hausdorff distance
Q: How close is Amoba to ArchTrop? Hausdorff distance - measure of
closeness:
∆X, Y = min{sup inf d(x, y), sup inf d(x, y)}
x∈X y∈Y
Roman Kogan
y∈Y x∈X
Amoebas, ArchTrop and all that jazz
March 24, 2014
13 / 18
Hausdorff distance
Q: How close is Amoba to ArchTrop? Hausdorff distance - measure of
closeness:
∆X, Y = min{sup inf d(x, y), sup inf d(x, y)}
x∈X y∈Y
Roman Kogan
y∈Y x∈X
Amoebas, ArchTrop and all that jazz
March 24, 2014
13 / 18
Hausdorff distance
Q: How close is Amoba to ArchTrop? Hausdorff distance - measure of
closeness:
∆X, Y = min{sup inf d(x, y), sup inf d(x, y)}
x∈X y∈Y
y∈Y x∈X
X
Y
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
13 / 18
Hausdorff distance
Q: How close is Amoba to ArchTrop? Hausdorff distance - measure of
closeness:
∆X, Y = min{sup inf d(x, y), sup inf d(x, y)}
x∈X y∈Y
y∈Y x∈X
X
Y
E.g. above:
supx∈X inf y∈Y d(x, y) is small: X close to Y ;
supy∈Y inf x∈X d(x, y) is big: Y not close to X.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
13 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Main Theorem
Theorem
For all p with τ terms, ∆(Amoeba(p), ArchTrop(p) ≤ (2τ − 3) log(τ − 1))
Theorem
Furthermore:
if vertices of support of p = vertices of Newt(p), then
ArchTrop(p) ⊂ Amoeba(p) and they are homotopy equivalent;
supx∈Amoeba inf y∈ArchTrop d(x, y) ≤ log(τ − 1), so there is hope for
improvement!
For p(x1 , . . . , xk ) = (x1 + 1)τ −k + x2 + . . . + xk ,
∆(Amoeba(p), ArchTrop(p) ≥ log(τ − k)).
These bounds in number of of terms only are new.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
14 / 18
Complexity results
Complexity is in terms of problem bitsize.
Deciding whether 0 ∈ Amoeba(p), p ∈ Z[x] is NP-hard [Plaisted, 84];
For fixed n, if f has rational logs, computing Archtrop(p) is in P.
For fixed n, if f has rational coefficients, deciding q ∈ Archtrop(p) is
in P.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
15 / 18
Complexity results
Complexity is in terms of problem bitsize.
Deciding whether 0 ∈ Amoeba(p), p ∈ Z[x] is NP-hard [Plaisted, 84];
For fixed n, if f has rational logs, computing Archtrop(p) is in P.
For fixed n, if f has rational coefficients, deciding q ∈ Archtrop(p) is
in P.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
15 / 18
Complexity results
Complexity is in terms of problem bitsize.
Deciding whether 0 ∈ Amoeba(p), p ∈ Z[x] is NP-hard [Plaisted, 84];
For fixed n, if f has rational logs, computing Archtrop(p) is in P.
For fixed n, if f has rational coefficients, deciding q ∈ Archtrop(p) is
in P.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
15 / 18
Complexity results
Complexity is in terms of problem bitsize.
Deciding whether 0 ∈ Amoeba(p), p ∈ Z[x] is NP-hard [Plaisted, 84];
For fixed n, if f has rational logs, computing Archtrop(p) is in P.
For fixed n, if f has rational coefficients, deciding q ∈ Archtrop(p) is
in P.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
15 / 18
Gnarly details
Proof outline:
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Use basic properties to reduce to a special case;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Use basic properties to reduce to a special case;
Use elementary and classical results to show that
sup
inf
x∈Archtrop y∈Amoeba
Roman Kogan
d(x, y) ≤ 1 + log(τ − 1)
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Use basic properties to reduce to a special case;
Use elementary and classical results to show that
sup
inf
x∈Archtrop y∈Amoeba
d(x, y) ≤ 1 + log(τ − 1)
In many dimensions, specialize to one to generalize.
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Use basic properties to reduce to a special case;
Use elementary and classical results to show that
sup
inf
x∈Archtrop y∈Amoeba
d(x, y) ≤ 1 + log(τ − 1)
In many dimensions, specialize to one to generalize.Use inequalities
and convexity to find q ∈ Archtrop near a p ∈ Amoeba;
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Proof outline:
One-dimensional case:
Use basic properties to reduce to a special case;
Use elementary and classical results to show that
sup
inf
x∈Archtrop y∈Amoeba
d(x, y) ≤ 1 + log(τ − 1)
In many dimensions, specialize to one to generalize.Use inequalities
and convexity to find q ∈ Archtrop near a p ∈ Amoeba; Use results of
Passare and Rullgard to show that Amoeba is homotopic to the spine
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
16 / 18
Gnarly details
Showing supx∈Archtrop inf y∈Amoeba d(x, y) ≤ 1 + log(τ − 1) :
1
if g(x) = αxβ (γ1 x1 , γ2 x2 , . . . , γn xn ), then
Amoeba(g) = Amoeba(f ) − log |γ|
Archtrop(g) = Archtrop(f ) − log |γ|
2
Obtain a constant bound for p(x) of the form
p(x) = c0 + . . . + cd xd , |ci | ≤ 1, |c0 | = |cd | = 1;
3
Use a result of Montel to show the O(log |τ − 1|) bound for
p(x) = c0 + . . . + cp xp + γ1 xn1 + . . . + γq xnq ; |ci |, ||γi | ≤ 1
4
and |c0 | = |cd | = 1;
use observation 1 to generalize
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
17 / 18
Gnarly details
Showing supx∈Archtrop inf y∈Amoeba d(x, y) ≤ 1 + log(τ − 1) :
1
if g(x) = αxβ (γ1 x1 , γ2 x2 , . . . , γn xn ), then
Amoeba(g) = Amoeba(f ) − log |γ|
Archtrop(g) = Archtrop(f ) − log |γ|
2
Obtain a constant bound for p(x) of the form
p(x) = c0 + . . . + cd xd , |ci | ≤ 1, |c0 | = |cd | = 1;
3
Use a result of Montel to show the O(log |τ − 1|) bound for
p(x) = c0 + . . . + cp xp + γ1 xn1 + . . . + γq xnq ; |ci |, ||γi | ≤ 1
4
and |c0 | = |cd | = 1;
use observation 1 to generalize
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
17 / 18
Gnarly details
Showing supx∈Archtrop inf y∈Amoeba d(x, y) ≤ 1 + log(τ − 1) :
1
if g(x) = αxβ (γ1 x1 , γ2 x2 , . . . , γn xn ), then
Amoeba(g) = Amoeba(f ) − log |γ|
Archtrop(g) = Archtrop(f ) − log |γ|
2
Obtain a constant bound for p(x) of the form
p(x) = c0 + . . . + cd xd , |ci | ≤ 1, |c0 | = |cd | = 1;
3
Use a result of Montel to show the O(log |τ − 1|) bound for
p(x) = c0 + . . . + cp xp + γ1 xn1 + . . . + γq xnq ; |ci |, ||γi | ≤ 1
4
and |c0 | = |cd | = 1;
use observation 1 to generalize
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
17 / 18
Gnarly details
Showing supx∈Archtrop inf y∈Amoeba d(x, y) ≤ 1 + log(τ − 1) :
1
if g(x) = αxβ (γ1 x1 , γ2 x2 , . . . , γn xn ), then
Amoeba(g) = Amoeba(f ) − log |γ|
Archtrop(g) = Archtrop(f ) − log |γ|
2
Obtain a constant bound for p(x) of the form
p(x) = c0 + . . . + cd xd , |ci | ≤ 1, |c0 | = |cd | = 1;
3
Use a result of Montel to show the O(log |τ − 1|) bound for
p(x) = c0 + . . . + cp xp + γ1 xn1 + . . . + γq xnq ; |ci |, ||γi | ≤ 1
4
and |c0 | = |cd | = 1;
use observation 1 to generalize
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
17 / 18
Gnarly details
Showing supx∈Archtrop inf y∈Amoeba d(x, y) ≤ 1 + log(τ − 1) :
1
if g(x) = αxβ (γ1 x1 , γ2 x2 , . . . , γn xn ), then
Amoeba(g) = Amoeba(f ) − log |γ|
Archtrop(g) = Archtrop(f ) − log |γ|
2
Obtain a constant bound for p(x) of the form
p(x) = c0 + . . . + cd xd , |ci | ≤ 1, |c0 | = |cd | = 1;
3
Use a result of Montel to show the O(log |τ − 1|) bound for
p(x) = c0 + . . . + cp xp + γ1 xn1 + . . . + γq xnq ; |ci |, ||γi | ≤ 1
4
and |c0 | = |cd | = 1;
use observation 1 to generalize
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
17 / 18
Applications and further thoughts
This can be used to isolate roots;
Connection to triangulations and Voronoi can be explored;
Details in a paper [AKNR] Metric Estimates and Membership
Complexity for Archimedean Amoebae and Tropical Hypersurfaces
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
18 / 18
Applications and further thoughts
This can be used to isolate roots;
Connection to triangulations and Voronoi can be explored;
Details in a paper [AKNR] Metric Estimates and Membership
Complexity for Archimedean Amoebae and Tropical Hypersurfaces
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
18 / 18
Applications and further thoughts
This can be used to isolate roots;
Connection to triangulations and Voronoi can be explored;
Details in a paper [AKNR] Metric Estimates and Membership
Complexity for Archimedean Amoebae and Tropical Hypersurfaces
Roman Kogan
Amoebas, ArchTrop and all that jazz
March 24, 2014
18 / 18