Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis...

Geometry of R Roots of 9×9 PolynomialSystems

Luis FelicianoTexas A&M University

July 16, 2018REU: DMS – 1757872

Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION

Motivation

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

A Quadratic Pentanomial 2x2 system!

Solving polynomial equations becomes more and morecomplicated as we increase the number of terms and variables.

Today we’ll take a look at some constructions that give usrough approximations for roots in a fraction of the time!

Motivation

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

Motivation

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

A Quick Review

A convex set is a set of points such that, given any two pointsP ,Q , then the line segment PQ is also in the set

A Quick Review

A convex set is a set of points such that, given any two pointsP ,Q , then the line segment PQ is also in the set

A Quick Review

A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set

A Quick Review

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

conv{S} := smallest convex set containing S

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

Convex Hull

Denoted by conv{·}

For any S ⊂ Rn

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

where xai = xa1,i1 x

a2,i2 · · · xan,i

Newt(f) := conv{ai | ci 6= 0}

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Denoted by Newt(·)

f(x) =t∑

i=1cix

a2,i2 · · · xan,i

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Although we didn’t make use of the following in our research...

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Newton Polytope

Although we didn’t make use of the following in our research...

The volume of the Newton polytope can be used to computethe degree of the corresponding hypersurface, and via mixedvolumes, the number of roots of systems of equations!

Ex:f(x, y) = 1 + x2 + y3 − 100xy

= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

Archimedean Newton Polytope

Denoted by ArchNewt(f)

ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

conv{(0, 0,−Log(1))

, (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1

⇒ ArchNewt(f) =

⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}

Archimedean Tropical Variety

We need two things to construct ArchTrop(f)

→ The outer normals of ArchNewt(f) that point downwards

Let’s project the lower faces of ArchNewt(f) onto the xy-plane

This gives us a triangulation of our Newton Polytope!

We take the outer normals of these lower faces

→We normalize them to be of the form (w,−1), and take

w to be a vertex of ArchTrop(f)!

→We normalize them to be of the form (w,−1), and take

w to be a vertex of ArchTrop(f)!

→ Roughly* translates to a point in each triangle!

→ The outer normals of the edges of Newt(f)

→ The outer normals of the exterior edges of Newt(f)

Putting these two together, we get...

Archimedean Tropical Variety - Roughly*

The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)

The rays are dual to the edges of Newt(f)

ArchTrop(f) gives us metric information about the roots andareas where we can find constant isotopy types!

A Word on Isotopy Types

Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots

ArchTrop(f) can do this for more general curves

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

⇒ The zero set of f(x, y) is either ∅, a point, or an oval!

⇒ This occurs when c < 6

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)

, c = 6

, c > 6

, respectively

Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of rootsArchTrop(f) can do this for more general curvesEx:f(x, y) = 1 + x2 + y3 − cxy (c > 0)⇒ The zero set of f(x, y) is either ∅, a point, or an oval!⇒ This occurs when c < 6

, c = 6

, c > 6

, respectively

ArchTrop+(f)

ArchTrop+(f) ⊂ ArchTrop(f)

This time we focus on the signs of our coefficients!

f(x, y) = 1 + x2 + y3 − 100xy

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

More specifically, we are interested in alternating signs!

ArchTrop+(f)

f(x, y) = 1 + x2 + y3 − 100xy

More specifically, we are interested in alternating signs!

We go from this...

To this!

ArchTrop+(f) gives us a piecewise linear function thatresembles the set of positive roots

A Small Discrepancy...

f(x) = 1− 1.1x + x2

ArchTrop+(f) looks like

But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!

On the other hand, if you look at

{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

f(x) = 1− 1.1x + x2

= (0, 2)

Our Research

Our Research - Newton Polytope

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

Our Research - Newton Polytope

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1

Our Research - f1

ArchNewt(f1) given these coefficients is...

Our Research - f1

ArchNewt(f1) given these coefficients is...

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7

Our Research - f1

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7

Our Research

No matter the coefficients, these 5 cases encompass all thepossible triangulations of f1!

Our Research

No matter the coefficients, these 5 cases encompass all thepossible triangulations of f1!

Our Research

f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5

Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1

Our Research

f2(x8, x9) = c6x8x9 + c7x28 + c8x8 + c9x9 + c10

Suppose c6 = 1, c7 = −10, c8 = −10, c9 = 2, c10 = 1

Our Research

A Theorem on ArchTrop(f)

ZC(f) := the Complex zero set of f

Theorem

For any pentanomial f in C[x1, . . . , xn], any point of Log|ZC(f)|is within distance log(4) of some point of ArchTrop(f).

Theorem

A Theorem on ArchTrop+(f)

Z+(f) := the positive zero set of f

Theorem

For any pentanomial f in R[x1, . . . , xn], any point of Log|Z+(f)|is within distance log(4) of some point of ArchTrop+(f).

A Theorem on ArchTrop+(f)

Z+(f) := the positive zero set of f

Theorem

For any pentanomial f in R[x1, . . . , xn], any point of Log|Z+(f)|is within distance log(4) of some point of ArchTrop+(f).

Using the same coefficients...

Theorem

If F is a random real 2× 2 quadratic pentanomial system withsupports having Cayley embedding

[2 1 1 0 0 0 1 1 0 00 1 0 1 0 2 1 0 1 0

such that the coefficient vector (c1, . . . , c10) has each ci withmean 0, then with probability at least 41%, F has the samenumber of positive roots as the cardinality ofArchTrop(f1) ∩ArchTrop(f2).

Successes!

Failures...

Failures...crickets...

Failures...

But why?

Some intuition...

But why?

Some intuition...

But why?

Some intuition...

But why?

Some intuition...

But why?

Some intuition...

But why?

Some intuition...

A Theorem on the Intersections

Theorem

For any 2× 2 polynomial system non-degenerate F withsupports having Cayley embedding A, the number of nonzeroreal roots of F depends only on the completed signedA-discriminant chamber containing F .

A Theorem on the Intersections

Theorem

For any 2× 2 polynomial system non-degenerate F withsupports having Cayley embedding A, the number of nonzeroreal roots of F depends only on the completed signedA-discriminant chamber containing F .

That being said...

We can compute the Hausdorff distance betweenArchTrop(f1) ∩ArchTrop(f2) and Log|Z+(f1)| ∩ Log|Z+(f2)|for 1000 random examples to obtain the following:

That being said...

Future Research

1. Generalizing our code

2. Finding the conditions under which

h0(Z+(f)) = h0(ArchTrop+(f))

3. Stability and the Jacobian

Future Research

Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis...

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