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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!