Does tropical geometry look like…

No, it looks like …

“Life Math is pain, anyone who says differently is selling something”

• Goal: explain and prove some basic notions.• Key words: complex numbers, algebraic

curves, genus• Basic example: exactly one conic passes

through 5 points

Digression: polynomials

• Example: – roots

• Example:– roots

€

X 2 − 3X +2 = (X −1)(X −2) = 0

€

X =1,X = 2

€

X 2 +1= (X − i)(X + i) = 0

€

X = i,X = −i

(i2 = −1)

Digression: complex numbers

Digression: complex numbers

• Theorem: any polynomial of degree n has exactly n complex roots

€

X n + an−1Xn−1 + ...+ a1X + a0 = (X − z1)(X − z2)...(X − zn )

Algebraic curves

• Polynomials in two variables

• Degree

€

F = ai, jXiY j

i+ j≤n

∑

€

deg(ak,mXkY m ) = k +m

Algebraic curves

• Definition: real algebraic curve of degree n is the set of points

• Complex algebraic curve: the same but in

€

(x, y)∈ R2 ,F(x, y) = 0,degF = n

€

C2

x

y y-x2=0

Examples: lines

• Line

• Through any two different points only one line passes– Fix one point– Write equation on a,b,c

– The same for the second point

• Find the unique solution

€

aX +bY + c = 0

€

(x1, y1)

€

ax1 +by1 =−c

€

ax2 +by2 =−c

Examples: conics

• Conic

• There is only one conic passing through 5 points.– Because is a linear equation. €

aX 2 +bXY + cY 2 +dX + eY + f = 0

€

ax12 +bx1y1 + cy1

2 +dx1 + ey1 =− f

Real algebraic curves

3

x=3

F(x,y)=0

X

Y

Real algebraic curves

Complex algebraic curves

Complex algebraic curves = surfaces

The Earth = torus ?

Counting of algebraic curves

• How many lines pass through 1 point ?

• How many lines pass through 3 points?

• Theorem: Through 3d-1+g generic points the finite number of curves degree d and genus g pass.

How many curves?

• The simplest way to count – tropical geometry.

• In the tropical world things are simpler than in the complex world.

Hope you find your tropical world.

Thank you!