Topology of Positive Zero Sets of Bivariate Pentanomials Malachi - - PowerPoint PPT Presentation

Topology of Positive Zero Sets of Bivariate Pentanomials

Malachi Alexander1, Ashley De Luna2 Christian McRoberts3

1California State University, Monterey Bay 2California State Polytechnic University, Pomona 3Morehouse College

MSRI-UP 2017, 08/04/2017

Motivation 1876: Carl Gustav Axel Harnack investigated the number of connected components of real algebraic curves.

Motivation 1876: Carl Gustav Axel Harnack investigated the number of connected components of real algebraic curves. 1900: David Hilbert posed his sixteenth problem about determining arrangements of algebraic curves.

Motivation Previous work has shown, For n-variate (n + 1)-nomials:

OR

Motivation Previous work has shown, For n-variate (n + 1)-nomials:

OR

For n-variate (n + 2)-nomials:

OR

Motivation Previous work has shown, For n-variate (n + 1)-nomials:

OR

For n-variate (n + 2)-nomials:

OR

Our work is on n-variate (n + 3)-nomials, We focus on the first unknown case: bivariate pentanomials!

Motivation Automate computing the topology of positive zero sets. Graph the A-discriminant curve of sparse polynomials.

Motivation

1 2 3 4 6 12 11 5 7 10 8 9

Automate computing the topology of positive zero sets. Graph the A-discriminant curve of sparse polynomials. Observe number of connected components.

Motivation Automate computing the topology of positive zero sets. Graph the A-discriminant curve of sparse polynomials. Observe number of connected components. Visualize the topology of the positive zero sets.

Motivation Example 1: f(x1, x2) = x100

1

x100

2

+ 1 + x2001

1

+ x2011

2

− 10x2

1x2 2

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

Trop+(f)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Z+(f)

Motivation Example 2: f(x1, x2) = 1 + x1 + x2 − 6

5x4 1x2 − 6 5x1x4 2

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2

• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

Trop+(f)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Z+(f)

Project Overview Step 1: Plot the A-Discriminant Variety Step 2: Compute Positive Tropical Variety Step 3: Find the Topology of Inner Chambers

Step 1: Plot the A-Discriminant Variety A-Discriminant Variety Let A ∈ Z2×5, with columns a1, . . . , a5 and x = (x1, x2) s.t. f(x) = c1xa1 + c2xa2 + c3xa3 + c4xa4 + c5xa5 ∈ C[x1, x2] ∇A := {[c1 · · · c5] ∈ P4

C|f(x) has a degenerate root in (C\{0})2}

[c1, . . . , c5] ∈ ∇A [c1, . . . , c5] ∈ ∇A

Step 1: Plot the A-Discriminant Variety ∇A has simple (C\{0})2-fibers

• ver a two-dimensional base.

Step 1: Plot the A-Discriminant Variety ∇A has simple (C\{0})2-fibers

• ver a two-dimensional base.

We take a two-dimensional slice to get the reduced A-discriminant variety.

Step 1: Plot the A-Discriminant Variety Given any polynomial f with x = (x1, x2), ai = (a1i, a2i), f(x) = c1xa1 + c2xa2 + c3xa3 + c4xa4 + c5xa5

Step 1: Plot the A-Discriminant Variety Given any polynomial f with x = (x1, x2), ai = (a1i, a2i), f(x) = c1xa1 + c2xa2 + c3xa3 + c4xa4 + c5xa5

A =

• 1

−1 5 4 9 4 2 3 1 4

• A =

2 1 5 8 4 2 8 8 4 1

Step 1: Plot the A-Discriminant Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 1: Plot the A-Discriminant Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

Step 1: Plot the A-Discriminant Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• ˆ

A =   1 1 1 1 1 2 1 5 8 4 2 8 8 4 1  

Step 1: Plot the A-Discriminant Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• ˆ

A =   1 1 1 1 1 2 1 5 8 4 2 8 8 4 1   We define B to be the right null space of ˆ A: ˆ AB = O where O is the zero vector.

Step 1: Plot the A-Discriminant Variety Use the Horn-Kapranov Uniformization.

θ

π

Step 1: Plot the A-Discriminant Variety Use the Horn-Kapranov Uniformization. ϕA(λ) := (Log|λBT|)B

θ

π

Step 1: Plot the A-Discriminant Variety Use the Horn-Kapranov Uniformization. ϕA(λ) := (Log|λBT|)B λ = (λ1, λ2) λ = (sin(θ), cos(θ))

θ

π

Step 1: Plot the A-Discriminant Variety Use the Horn-Kapranov Uniformization. ϕA(λ) := (Log|λBT|)B λ = (λ1, λ2) λ = (sin(θ), cos(θ)) 0 ≤ θ < π

θ

π

Step 1: Plot the A-Discriminant Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Project Overview Step 1: Plot the A-Discriminant Variety Step 2: Compute Positive Tropical Variety Step 3: Find the Topology of Inner Chambers

Step 2: Compute the Positive Tropical Variety Theorem 1. If f ∈ R[x1, x2] has coefficient vector c and Log|c|B lying in an outer chamber, then Z+(f) and Trop+(f) are isotopic.

Step 2: Compute the Positive Tropical Variety Theorem 1. If f ∈ R[x1, x2] has coefficient vector c and Log|c|B lying in an outer chamber, then Z+(f) and Trop+(f) are isotopic. f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cf =

• 10

−8 −11 8 3

• g(x1, x2) = 2x2

1x2 2 − 5x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cg =

• 2

−5 −11 8 3

• h(x1, x2) = 11x2

1x2 2 − 8x1x8 2 − 10x5 1x8 2 + 7x8 1x4 2 + 2x4 1x2

ch =

• 11

−8 −10 7 2

Step 2: Compute the Positive Tropical Variety

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

Log|c|B = (α, β)

Step 2: Compute the Positive Tropical Variety

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

Log|c|B = (α, β) f(x1, x2)

Step 2: Compute the Positive Tropical Variety

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

Log|c|B = (α, β) f(x1, x2) g(x1, x2)

Step 2: Compute the Positive Tropical Variety

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

• 50
• 40
• 30
• 20
• 10

10 20 30 40 50

Log|c|B = (α, β) f(x1, x2) g(x1, x2) h(x1, x2)

Step 2: Compute the Positive Tropical Variety Theorem 1. If f ∈ R[x1, x2] has coefficient vector c and Log|c|B lying in an outer chamber, then Z+(f) and Trop+(f) are isotopic. f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cf =

• 10

−8 −11 8 3

• g(x1, x2) = 2x2

1x2 2 − 5x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cg =

• 2

−5 −11 8 3

• h(x1, x2) = 11x2

1x2 2 − 8x1x8 2 − 10x5 1x8 2 + 7x8 1x4 2 + 2x4 1x2

ch =

• 11

−8 −10 7 2

Step 2: Compute the Positive Tropical Variety Isotopy: Given two sets X, Y ∈ R2, a homeomorphism ι : [0, 1] × R2 → R2 is an isotopy if ι is a continuous map such that ι(0, X) = X and ι(1, X) = Y

Step 2: Compute the Positive Tropical Variety Isotopy: Given two sets X, Y ∈ R2, a homeomorphism ι : [0, 1] × R2 → R2 is an isotopy if ι is a continuous map such that ι(0, X) = X and ι(1, X) = Y

Step 2: Compute the Positive Tropical Variety Theorem 1. If f ∈ R[x1, x2] has coefficient vector c and Log|c|B lying in an outer chamber, then Z+(f) and Trop+(f) are isotopic. f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cf =

• 10

−8 −11 8 3

• g(x1, x2) = 2x2

1x2 2 − 5x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

cg =

• 2

−5 −11 8 3

• h(x1, x2) = 11x2

1x2 2 − 8x1x8 2 − 10x5 1x8 2 + 7x8 1x4 2 + 2x4 1x2

ch =

• 11

−8 −10 7 2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• Newt(f) := Conv({ai})

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• ArchNewt(f) := Conv({ai, −Log|ci|})
• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• ArchNewt(f) := Conv({ai, −Log|ci|})
• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• ArchNewt(f) := Conv({ai, −Log|ci|})
• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• ArchNewt(f) := Conv({ai, −Log|ci|})
• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

A = 2 1 5 8 4 2 8 8 4 1

• c =
• 10

−8 −11 8 3

• ArchNewt(f) := Conv({ai, −Log|ci|})
• 3

8

• 2

7

• 1

8 6 5 6 4 4 3 2 2 1

Step 2: Compute the Positive Tropical Variety

• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety

• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety

• 2.2
• 2
• 1.8
• 1.6

8

• 1.4
• 1.2

8 6 6 4 4 2 2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension} (4, 2, −0.5) → (8, 4, −1)

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, −1)

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• +

+ +

f(x1, x2) = 10x2

1x2 2−8x1x8 2−11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• +

+ +

f(x1, x2) = 10x2

1x2 2−8x1x8 2−11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• +

+ +

f(x1, x2) = 10x2

1x2 2−8x1x8 2−11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• +

+ +

f(x1, x2) = 10x2

1x2 2−8x1x8 2−11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Trop+(f) := {v ∈ Rn | (v, −1) is an outernormal of ArchNewt(f) with cicj < 0 for some face of positive dimension}

✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

(4, 2, −0.5) → (8, 4, ✟

✟ ❍ ❍

−1)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• +

+ +

f(x1, x2) = 10x2

1x2 2−8x1x8 2−11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

Step 2: Compute the Positive Tropical Variety Theorem 2. Suppose f, h ∈ R[x1, x2] have coefficient vectors cf and ch, respectively, with the same sign vectors, and Log|cf|B and Log|ch|B lying in the same reduced signed discriminant chamber, then the positive zero sets of f and h are isotopic.

Step 2: Compute the Positive Tropical Variety Theorem 2. Suppose f, h ∈ R[x1, x2] have coefficient vectors cf and ch, respectively, with the same sign vectors, and Log|cf|B and Log|ch|B lying in the same reduced signed discriminant chamber, then the positive zero sets of f and h are isotopic. f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

h(x1, x2) = 11x2

1x2 2 − 8x1x8 2 − 10x5 1x8 2 + 7x8 1x4 2 + 2x4 1x2

Step 2: Compute the Positive Tropical Variety Theorem 2. Suppose f, h ∈ R[x1, x2] have coefficient vectors cf and ch, respectively, with the same sign vectors, and Log|cf|B and Log|ch|B lying in the same reduced signed discriminant chamber, then the positive zero sets of f and h are isotopic. f(x1, x2) = 10x2

1x2 2 − 8x1x8 2 − 11x5 1x8 2 + 8x8 1x4 2 + 3x4 1x2

h(x1, x2) = 11x2

1x2 2 − 8x1x8 2 − 10x5 1x8 2 + 7x8 1x4 2 + 2x4 1x2

f(x1, x2) h(x1, x2) cf =

• 10

−8 −11 8 3

• ch =
• 11

−8 −10 7 2

• signf =
• +

− − + +

• signh =
• +

− − + +

Step 2: Compute the Positive Tropical Variety f(x1, x2)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

h(x1, x2)

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

Our Algorithm Input coefficient vector and exponent matrix Graph the relative discriminant curve

Our Algorithm Input coefficient vector and exponent matrix Graph the relative discriminant curve Create the 3D Archimedean Newton polygon

Illustrate outer normals of each facet

Create the 2D Newton polygon

Plot the (x,y) of the lower hulls outer normal alongside the polygon

Construct the polynomial’s zero set topology

Future Work Find Topologies of the Inner Chambers

Construct the ray extending from the origin Ray intersections with A-discriminant curve Solving transcendental equation Can be solved quickly (linear combination of logs)

Project Overview Step 1: Plot the A-Discriminant Variety Step 2: Compute Positive Tropical Variety Step 3: Find the Topology of Inner Chambers

Project Overview

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

Step 3: Find the Topology of Inner Chambers

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10 20 30 40 50

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10 20 30 40 50

By Horn-Kapranov Uniformization, we can solve αiLog(λi + βi) = 0 Example: 2log(t − 1) + 3log(2t + 3) + 5log(7) = 0

Acknowledgements We would like to thank Mathematical Sciences Research Institute (MSRI) National Science Foundation Grant DMS-1659138 Sloan Foundation Grant

• Dr. Federico Ardila, Dr. J. Maurice Rojas, Dr. Anastasia

Chavez, Megan Ly, Robert Walker Our fellow colleagues