Download Tropical geometry as is

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: X 2  3X  2  (X 1)(X  2)  0
– roots
X 1, X  2

• Example: X 2 1  (X  i)(X  i)  0
– roots
X  i, X  i
 (i  1)
2
Digression: complex numbers
Digression: complex numbers
• Theorem: any polynomial of degree n has
exactly n complex roots
X  an 1 X
n
n 1
 ... a1 X  a0  (X  z1 )(X  z2 )...( X  zn )
Algebraic curves
• Polynomials in two variables
F
a
i
i,j
XY
j
i jn
• Degree
deg(ak,m X Y )  k  m
k
m
Algebraic curves
• Definition: real algebraic curve of degree n is
the set of points
(x, y)  R , F(x, y)  0, deg F  n
2
• Complex algebraic curve: the same but in
y
y-x2=0
x

C
2

Examples: lines
• Line
aX bY  c  0
• Through any two different points only one line passes
– Fix one point (x , y )
1 1
– Write equation on a,b,c
1 by1  c
– The same for the secondax
point
• Find
the unique solution ax2 by2  c


Examples: conics
• Conic
aX  bXY  cY  dX  eY  f  0
2
2
• There is only one conic passing through 5
points.
– Because ax12 bx1y1  cy12  dx1  ey1   f
is a linear equation.

Real algebraic curves
Y
x=3
F(x,y)=0
3
X
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!