tropical algebraic geometry -...

Tropical Algebraic Geometry1 Amoebas

an asymptotic view on varietiestropicalization of a polynomial

2 The Max-Plus Algebra at Workmaking time tables for a railway network

3 Polyhedral Methods for Algebraic Setssolving the cyclic 4-roots problemasymptotics of witness sets

4 Tropical Polynomialsthe tropical fundamental theorem of algebragraphing a tropical line

MCS 563 Lecture 39Analytic Symbolic ComputationJan Verschelde, 21 April 2014

Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 1 / 34

Tropical Algebraic Geometry

1 Amoebasan asymptotic view on varietiestropicalization of a polynomial





logarithms of algebraic sets

Denoting C∗ = C \ {0}, we consider the application of

log : C∗ × C

∗ → R × R

(x , y) 7→ (log(|x |), log(|y |))

to a variety.

Introduced by Gelfand, Kapranov, and Zelevinsky,log(V ) for a variety V is called the amoeba of a variety.


plot for a line

For a line, we use polar coordinates to plot the amoeba:f := 1

2x + 15y − 1 = 0, A :=

[

ln(∣

∣reIθ∣

∣

)

, ln(∣

∣

52 reIθ − 5

∣

∣

)]

.

> f := 1/2*x + 1/5*y - 1:> s := solve(f,y):> L := map(log,map(abs,[x,s])):> A := subs(x=r*exp(I*theta),L);> Ap := seq(plot([op(subs(theta=k*Pi/200,A)),r=-100..100],thickness=6),k=0..99):

> plots[display](Ap,axes=none);

The amoeba of a linear polynomial with Maple.


compactifying the amoeba

We compactify the amoeba of f−1(0):

Take lines perpendicular to the tentacles.

As each line cuts the plane in half, keep those halves of the planewhere the amoeba lives.

The intersection of all half planes defines a polygon.

The resulting polygon is the Newton polygon of f .


amoeba and Newton polygon

The edges of the Newton polygon are perpendicularto the tentacles of the amoeba.

K2 0 2 4 6 8 10 12

K2

2

4

6

8

10

12

t(0,0)

t(1,0)

t(0,1)

@@

@@


normal cones and fan

The inner product is〈·, ·〉 : Z

2 × Z2 → Z : ((i , j), (u, v)) 7→ iu + jv .

Let P be the Newton polygon of the polynomial f .The normal cone to a vertex p of P is

{ v ∈ R2 \ {0} | 〈p, v〉 = min

q∈P〈q, v〉 }.

The normal cone to an edge spanned by p1 and p2 is

{ v ∈ R2 \ {0} | 〈p1, v〉 = 〈p2, v〉 = min

q∈P〈q, v〉 }.

The normal fan of P is the collection of all normal cones to verticesand edges of P.Given f , a tropicalization of f , denoted by Trop(f ), is a finite collectionof inner normals (u, v) (normalized as gcd(u, v) = 1) to the edges ofthe Newton polygon P of f .


a quadratic polynomial

Consider a regular triangulation of the Newton polygon of a generalquadratic polynomial.

Take the normals to the facets of the upper hull:

rr@

@@rr

r

r

rr

rr

��

��

�

��

��


intersecting two quadrics

Consider two general quadratic polynomials:

rr

rr

��

��

��

�

��

��

��

rr

rr

��

��

rr@

@@rr

r

rrr@

@@rr

r

r

We see a tropical version of the theorem of Bézout.

This tropical version is an interpretation of the computation of themixed cells in a regular mixed-cell configuration via the commonrefinement of the normal fans to their edges.


a dictionary

An analogy between classical algebraic geometryand tropical algebraic geometry:

classical ↔ tropical

C,+, · R, min,+

polynomial piecewise linear functionzero set singular locus

hypersurface normal fantoric variety ordinary linear variety

irreducible algebraic variety balanced (n − 1)-skeleton


a railway networkConsider two stations: S1 and S2; three circuits: one inner between thetwo stations and two outer circuits, serving the suburbs. Four trainstravel back and forth.

S1 S2

? ?6 62 3

3

5

Numbers along the tracks are fixed travel times.A good timetable obey the rules:

1 the frequency of trains should be as high as possible;2 the frequency of trains is the same along all tracks;3 trains wait to allow the changeover of passengers;4 the trains depart at a station as soon as allowed.


inequalities

Denote by x1 and x2 the common departure times of the trainsrespectively at S1 and S2.

The initial departure times are x1(0) and x2(0) andthe k th departure times are given by x1(k − 1) and x2(k − 1).

The rules above translate into the inequalities:

x1(k + 1) = max(x1(k) + 2, x2(k) + 5)x2(k + 1) = max(x1(k) + 3, x2(k) + 3)

In the max-plus algebra,replacing max by ⊕ and addition by ⊗, we write

x1(k + 1) = (x1(k) ⊗ 2) ⊕ (x2(k) ⊗ 5)x2(k + 1) = (x1(k) ⊗ 3) ⊕ (x2(k) ⊗ 3)


tropical linear algebra

In matrix-vector form:

x(k + 1) = A ⊗ x(k), x(k) =

(

x1(k)x2(k)

)

, A =

(

2 53 3

)

.

Eigenvalues λ and eigenvectors v of a matrix A are defined in themax-plus algebra as

A ⊗ v = λ ⊗ v,

where not all components of v are equal to −∞.


tropical eigenvalues

Eigenvectors are (similar to conventional linear algebra)not uniquely determined.

If the same constant is added to all components of an eigenvector,then we get again an eigenvector.

It is typical to normalize an eigenvector so its first component is zero.

For the solution of the system above, we write

x(1) = A ⊗ x(0) = λ ⊗ x(0)

and in generalx(k) = λ⊗k ⊗ x(0), k = 1, 2, . . .


curves of cyclic 4-roots

f (x) =

x0 + x1 + x2 + x3 = 0x0x1 + x1x2 + x2x3 + x3x0 = 0

x0x1x2 + x1x2x3 + x2x3x0 + x3x0x1 = 0x0x1x2x3 − 1 = 0

One tropism v = (+1,−1,+1,−1) with inv(f )(z) = 0:

inv(f )(x) =

x1 + x3 = 0x0x1 + x1x2 + x2x3 + x3x0 = 0

x1x2x3 + x3x0x1 = 0x0x1x2x3 − 1 = 0.

We look for solutions of the form

(x0 = t+1, x1 = z1t−1, x2 = z2t+1, x3 = z3t−1).


solving the initial form system

Substitute (x0 = t+1, x1 = z1t−1, x2 = z2t+1, x3 = z3t−1):

inv(f )(x0 = t+1, x1 = z1t−1, x2 = z2t+1, x3 = z3t−1)

=

(1 + z2)t+1 = 0z1 + z1z2 + z2z3 + z3 = 0

(z1z2 + z3z1)t+1 = 0z1z2z3 − 1 = 0.

We find two solutions: (z1 = −1, z2 = −1, z3 = +1)and (z1 = +1, z2 = −1, z3 = −1).

Two space curves(

t ,−t−1,−t , t−1)

and(

t , t−1,−t ,−t−1)

satisfy the entire cyclic 4-roots system.


Asymptotics of Witness SetsConsider a witness sets for an algebraic space curve.

A witness set for a degree d curve contains d points on a generalhyperplane c0 + c1x1 + c2x2 + · · · + cnxn = 0, satisfying f (x) = 0.

We then deform a witness set for a curve in two stages:1 The first homotopy moves to a hyperplane in special position:

h(1)(x, t) =

{

f (x) = 0, for t from 1 to 0 :(c0 + c1x1 + · · · + cnxn)t + (c0 + c1x1)(1 − t) = 0.

2 After renaming c0 + c1x1 = 0 into x1 = γ, we let x1 go to zero withthe following homotopy:

h(2)(x, t) =

{

f (x) = 0x1 − γt = 0, for t from 1 to 0.

The two homotopies need further study.


the first homotopy

In moving a general plane into special position, some paths willdiverge, consider for example f (x1, x2) = x1x2 − 1.

Lemma

All solutions at the end of the homotopy h(1)(x, t) = 0 lie on the curvedefined by f (x) = 0 and in the hyperplane x1 = −c0/c1.

Proof. We claim that we find the same solutions to h(1)(x, t = 0) = 0either by using the homotopy

h(1)(x, t) =

{

f (x) = 0, for t from 1 to 0 :(c0 + c1x1 + · · · + cnxn)t + (c0 + c1x1)(1 − t) = 0.

or by solving h(1)(x, 0) = 0 directly. This claim follows from cheater’shomotopy or the more general coefficient-parameter polynomialcontinuation.


the second homotopyFor simplest example of the hyperbola x1x2 − 1 = 0: its solutionis (x1 = t , x2 = t−1) and the tropism is v = (1,−1).

Lemma

As t → 0 in the homotopy h(2)(x, t) = 0, the leading powers of thePuiseux series expansions are the components of a tropism.In particular, the expansions have the form

{

x1 = txk = ck tvk (1 + O(t)), k = 2, . . . , n.

(1)

Proof. Following Bernshteı̌n’s second theorem, a solution at infinity is asolution in (C∗)n of an initial form system. For a solution to have valuesin (C∗)n, all equations in that system need to have at least twomonomials. So the system is an initial form system defined by atropism. To arrive at the form of (1) for the solution of h(2)(x, t) = 0 werescale the parameter t so we may replace x1 = γt by x1 = t .


tropical monomials

Let x1, x2, . . . , xn be variables representing elements in(R ∪ {∞},⊕,⊙), then a tropical monomial is a product of variables,allowing repetition. Example:

x2 ⊙ x1 ⊙ x3 ⊙ x1 ⊙ x4 ⊙ x2 ⊙ x3 ⊙ x2 = x⊙21 x⊙3

2 x⊙23 x4.

As a function, a tropical monomial is a linear function.

x2 + x1 + x3 + x1 + x4 + x2 + x3 + x2 = 2x1 + 3x2 + 2x3 + x4.

Every linear function in n variables with integer coefficientswe can write as a tropical monomial, exponents can be negative.

Tropical monomials are the linear functions on Rn

with integer coefficients.


tropical polynomials

Any finite linear combination of tropical monomials defines a tropicalpolynomial with real coefficients and integer exponents:

p(x1, x2, . . . , xn) = ca1 ⊙ x⊙a1,11 x

⊙a1,22 · · · x

⊙a1,nn

⊕ ca2 ⊙ x⊙a2,11 x

⊙a2,22 · · · x

⊙a2,nn

⊕ · · ·

⊕ cak ⊙ x⊙ak,11 x

⊙ak,22 · · · x

⊙ak,nn .

The corresponding function:

p(x1, x2, . . . , xn) = min( ca1 + a1,1x1 + a1,2x2 + · · · + a1,nxn,

ca2 + a2,1x1 + a2,2x2 + · · · + a2,nxn,

. . . ,

cak + ak ,1x1 + ak ,2x2 + · · · + ak ,nxn ).


piecewise-linear functions

Tropical polynomials as functions p : Rn → R

1 are continuous;2 are piecewise-linear, with a finite number of pieces; and3 are concave, i.e.: p

(x+y2

)

≥ 12(p(x) + p(y)) for all x , y ∈ R.

Any function which satisfies these three properties can be representedas the minimum of a finite set of linear functions.

LemmaThe tropical polynomials in n variables are the piecewise-linearfunctions on R

n with integer coefficients.


the graph of a tropical quadratic polynomial

a ⊙ x⊙2 ⊕ b ⊙ x ⊕ c = min(a + 2x , b + x , c)

��

��

��

��

��

��

��

� b + x

��a + 2x

cg

c − b

gb − a

� @@I

roots

��

��

��

��

��

��

��

��

��

�

��

��

�

If b − a ≤ c − b, then p(x) = a ⊙ (x ⊕ (b − a)) ⊙ (x ⊕ (c − b))= a + min(x , b − a) + min(x , c − b).


the tropical fundamental theorem of algebra

TheoremEvery tropical polynomial function can be written uniquelyas a tropical product of tropical linear functions.

Note:

Distinct polynomials can represent the same function.

6

-��

1 + x

��

��

2x

2

x2 ⊕ 17 ⊙ x ⊕ 2 = min(2x , x + 17, 2)

= min(2x , x + 1, 2)

= x2 ⊕ 1 ⊙ x ⊕ 2= (x ⊕ 1)⊙2

Every polynomial can be replaced by an equivalent polynomial(equivalent means: representing the same function)that can be factored into linear functions.


graph of the tropical line L(x , y) = 1 ⊙ x ⊕ 2 ⊙ y ⊕ 3

��1 yB

BBBB

BB

BBB

BBB

BBB

BBMx

�� (2,1)

(2,1,3)�� x = 2

��

��

��

��

��

��

��

BBB

BBB

BBy = 1

BBB

BBBB

BB

BB

BBB

BB

BBB

BB��

BB

BBB

BB6z

(−1,−2)BB

BBB

BBB

BBB

BB

y = −2

��

��y = x − 1

��

��

��

��

�� x = −1��

��

L : R2 → R

(x , y) 7→ min(1 + x , 2 + y , 3)

z = 1 + xz = 2 + y

}

: y = x − 1

z = 1 + xz = 3

}

: x = 2

z = 2 + yz = 3

}

: y = 1


a picture in the plane

L : R2 → R : (x , y) 7→ min(1 + x , 2 + y , 3)

Look where the minimum is attained at least twice:

-x

6y

y = 1

x = 2

��

�y = x − 1

��

�

��

�


intersecting two tropical lines

Intersecting 1 ⊙ x ⊕ 2 ⊙ y ⊕ 3 with −2 ⊙ x ⊕ 4 ⊙ y ⊕ 2: wheremin(1+x , 2+y , 3) and min(−2+x , 4+y , 2) attain their mininum twice.

-x

6y

y = 1

x = 2

��

�y = x − 1

��

�

��

�

x = 4

y = −2�

��

y = x − 6

��

�

��

�

g


Summary + Exercises

Tropical algebraic geometry provides a framework to apply polyhedralmethods to solve polynomial systems.

Exercises:

1 Consider the amoeba of the product of two linear equations. Howmany tentacles to you see in each direction? For which choices ofthe coefficients do you see holes in the amoeba?

2 Prove the tropical fundamental theorem of algebra.3 Investigate the complexity of graphing a tropical plane curve of

degree d and express the complexity as a function of d .


tropical algebraic geometry -...

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