An Introduction to Tropical Mathematics

Turning Polynomials into Polygons

By: Sera Gunn (she/they)

Classical Algebraic Geometry

Ingredients

  • Polynomials \(f(x, y) = x^2 + y^2 - 1\)
  • Yielding algebraic curves: \(f(x, y) = 0\)
    • or surfaces: \(x^2 + y^2 + z^2 = 1\)
    • General terminology is “a variety” \[V(f) = \{(x, y) : f(x, y) = 0\}\]
  • Different number systems (real numbers \(\mathbf{R}\), complex numbers \(\mathbf{C}\), modular arithmetic \(\mathbf{F}_p\))

Algebraic Curves

The Unit Circle

Classical Results

Theorem 1 (Bézout’s Theorem) If \(f(x, y)\) has degree \(p\) and \(g(x, y)\) has degree \(q\) then generally the system \(f(x, y) = 0\), \(g(x, y) = 0\) has \(pq\) intersections.

Quadratic and Cubic

The circle (degree 2) and cubic (degree 3) have 6 intersection points

Tropicalization

Early Origins

1971: George Bergman The logarithmic limit-set of an algebraic variety

  • Take the (complex) points defined by \(f(x, y) = 0\)
  • Map every point \((x, y)\) to \((\log_b |x|, \log_b |y|)\)
  • Take the limit of this set as \(b \to \infty\)
    • … limit of a set?
    • just go with it…

A Tropical Curve

A Tropical Line

Primes and Valued Fields

Let \(p\) be a prime number.

Definition 1 For integers: \[ v_p(n) = \text{the largest power of $p$ that goes into $n$} \]

Definition 2 For rationals: \[ v_p(m/n) = v_p(m) - v_p(n) \]

Example 1 \[ v_5(200/125) = v_5(5^2 \cdot 2^3) - v_5(5^3) = 2 - 3 = -1 \]

Arithmetic

  1. Same value no matter how we express a fraction: \(v_5(200/125) = v_5(8/5)\)
  2. \(v_p(x \cdot y) = v_p(x) + v_p(y)\)
  3. \(v_p(x / y) = v_p(x) - v_p(y)\)
  4. \(v_p(x^k) = kv_p(x)\)

Remark 1. They behave like logarithms

Min-property

\[v_p(x + y) \ge \min\{v_p(x), v_p(y)\}\]

  • e.g. \[\begin{align*} v_5(250 + 200) &= v_5(5^3 \cdot 2 + 5^2 \cdot 2^3) \\ &= v_5(5^2(5 \cdot 2 + 2^3)) = 2 \end{align*}\]
  • More precisely: if \(v_p(x) \neq v_p(y)\) then \(v_p(x + y) = \min\{v_p(x), v_p(y)\}\)
  • If \(v_p(x) = v_p(y)\) then \(v_p(x + y)\) can be any value between \(v_p(x)\) and \(\infty\) (as long as \(p \neq 2\))
    • Wait, what happens when \(p = 2\)?
    • Investigate yourself!
    • e.g. \(v_2(12) = v_2(20) = 2\) but \(v_2(32) = 3\)

Equations over valued fields

\[x + y - 1 = 0\]

How do \(v_p(x)\) and \(v_p(y)\) relate?

  • If the minimum of \(v_p(x), v_p(y), v_p(-1) = 0\) is unique
    • \(v_p(x + y - 1) = \min\{v_p(x), v_p(y), 0\}\)
    • \(v_p(x + y - 1) \neq v_p(0) = \infty\)
    • \(x + y - 1 \neq 0\)
  • So the minimum is not unique
  • It occurs at least twice

“Curves” over valued fields

Start with an equation:

\[V(x + y - 1) = \{x + y - 1 = 0\}\]

Tropicalize:

\[V(\bar{x} \oplus \bar{y} \oplus 0) = \{\text{min occurs at least twice}\} \]

Notation: \(V\) means vanishing set or variety, \(\oplus = \min\).

Minimum occurs twice

\[\min\{\bar x, \bar y, 0\}\]

Cases:

  • \(\min = \bar x = \bar y \le 0\)
  • \(\min = \bar x = 0 \le \bar y\)
  • \(\min = \bar y = 0 \le \bar x\)

Min-convention Tropical Geometry

…And it’s upside down

The two conventions

  1. Min-convention \(v_p, \min, v_p(0) = \infty\)
  2. Max-convention \(|x|_p = p^{-v_p(x)}, \max, |0|_p = - \infty\)
    • exponential can have any base but \(p\) is the convention

Triangles

How to compute the tropical curve associated to a tropical polynomial?

\[ f(x,y) = 75 + 10x - 12y + 25xy \]

Record the valuation \(v_5\) at each monomial:

\[ \mathrm{trop}(f) = 2 \oplus \bar 1x \oplus \bar y \oplus 2 \bar x \bar y \]

Recipe

  • For each term \(a\bar x^m \bar y^n\) lift the point \((m, n)\) up to a height of \(a\).
    • I.e. \((m, n, a)\)
  • Wrap those points on the bottom with a sheet (lower convex hull).
  • Use the folds in the sheet to draw polygons between the points \((m, n)\).
  • The tropical curve is the dual graph of this complex.
    • Rotate 180 degrees for min-convention

Visualization

A tropical elliptic curve

A degree 3/genus 1 tropical curve

Tropical Bézout

Bézout’s theorem for tropical curves: a degree 3 and degree 2 tropical curve intersect in 6 points generically

Mixed Subdivisions

Cells corresponding to intersections are quadrangles. Multiplicity of intersection = area of quadrangle

Part II: Points on Curves

Elliptic Curves

A smooth cubic curve is called an elliptic curve

Weierstraß form \[ y^2 = x^3 + px + q \]

As complex surfaces these are toruses (genus 1)

Abstract Tropical Curves

An tropical curve

Abstract Tropical Curves

An abstract tropical curve

Abstract Tropical Curves

An abstract tropical curve (skeleton)

Source

  • Imagine we have an equation \(f(x,y)\) with integer coefficients.
  • We can reduce mod \(p\) to curves over the mod-\(p\) number system.
  • If we have a good equation, i.e. a bad equation, this reduced equation will have self-intersections and intersecting components
    • components correspond to points
    • intersections correspond to connections

Dollars on Graphs

Points on an abstract tropical curve

  • For a classical curve: \(P + Q + R = 0\) if they are colinear
  • For a tropical curve we have borrowing moves

Dollars on Graphs

Borrowing/Lending Move

  • At any point, a vertex may send 1 dollar simultaneously to each of its neighbours
  • Question: can we get everyone out of debt?

Example 1

Can you win the dollar game?

Example 2

Can you win the dollar game?

Example 3

Can you win the dollar game?

Rank Function

Let \(D\) represent a given dollar game.

  • If \(D\) is unwinnable, define \(r(D) = -1\)
  • If \(D\) is only just winnable, i.e. \(D\) is winnable but if we take a dollar away from \(v\), \(D - v\) is unwinnable, define \(r(D) = 0\)
  • Say \(r(D) \ge k\) if taking away any \(k\) dollars anywhere leaves a winnable game
  • \(r(D)\) will be the maximum \(k\)

Degree Function

\[\deg(D) = \text{total dollar amount of } D\]

Riemann-Roch Theorem

If a graph has genus \(g\) then for any game \(D\)

\[r(D) \ge \deg(D) + 1 - g\]

Classical Riemann-Roch

If a surface has genus \(g\), and \(D\) is a finite set of points with multiplicities. Consider the set of rational functions whose poles are not worse than the numbers specified in \(D\).

E.g. \[f = \frac{(x + 2)^4(x - 3)}{(x + 5)^3(x - 7)}\]

\[\mathrm(f) = \{4 \text{ at } -2, 1 \text{ at } 3, -3 \text{ at } -5, -1 \text{ at } 7\}\]

This is ok for \(D\) = no worse than \(-3\) at \(-5\) and no worse than \(-2\) at \(7\).

Classical Riemann-Roch (cont.)

A sum of two rational functions which is ok for \(D\) is also ok for \(D\). Likewise any constant multiple is ok for \(D\).

I.e. it is a vector space

Let \(\ell(D)\) be the dimension of this space.

Theorem 2 (Riemann-Roch) \[\ell(D) \ge \deg(D) + 1 - g\]