Tropical Extensions and Baker-Lorscheid Multiplicities

Trevor Gunn

November, 2022

$$\begin{align}\DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\In}{in} \DeclareMathOperator{\mult}{mult} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\mathbf{F}} \newcommand{\K}{\mathbf{K}} \newcommand{\N}{\mathbf{N}} \renewcommand{\P}{\mathbf{P}} \newcommand{\R}{\mathbf{R}} \renewcommand{\S}{\mathbf{S}} \newcommand{\T}{\mathbf{T}} \newcommand{\TC}{\mathbf{T\!C}} \newcommand{\TR}{\mathbf{T\!\!R}} \newcommand{\Z}{\mathbf{Z}}\end{align}$$


An idyll \(B = (B^\bullet, N_B)\) is

Such that \(B\) is field-like


$$\begin{align}\N[B^\bullet] = \{\text{finitely supported formal sums} \sum x_i\} / \langle 0_B \rangle\end{align}$$

Alternative notation:

$$\begin{align}\sum x_i \in N_B \longleftrightarrow 0 \leqslant \sum x_i\end{align}$$

Ordered Blueprints

A subaddition \(\leqslant\) is an additive and multiplicative preorder on \(\N[B^\bullet]\):

Idyllic Blueprints

\(B\) is idyllic if \(\leqslant\) is generated by relations of the form \(0 \leqslant x\)

An idyll is a field-like idyllic blueprint


\(\F_{1^n}\) has \(\F_{1^n}^\bullet = \{e^{2\pi i k/n} : k \in \Z\}\) and \(N_{\F_{1^n}} = \{\sum \theta_i : \text{sum is } 0 \in \C\}\)

\(\S\) (sign idyll) has \(\S^\bullet = \{0, 1, -1\}\) and \(N_{\S} = \{a \cdot 1 + b \cdot (-1) : ab \neq 0\}\)

\(\T\) (tropical idyll) has \(\T^\bullet = (\R \cup \{\infty\}, 0_\T = \infty, 1_\T = 0_\R, +)\) and \(\N_{\T} = \{\sum x_i : \text{min. occurs twice}\}\)

\(\K\) (Krasner idyll) is the subidyll of \(\T\) on \(\K^\bullet = \{0_\T, 1_\T\}\)

Polynomial Extensions

\(B[x]^\bullet = \{ bx^n : b \in B^\bullet, n \in \N \}\)

Whose subaddition is induced by that of \(B\)

A polynomial is \(\sum b_ix^i\) (at most one term in each degree)

Tropical Extensions

Let \(\Gamma\) be an ordered Abelian group (e.g. \(\R\))

\(B[\Gamma]^\bullet = \{ bx^\gamma : b \in B^\bullet, \gamma \in \Gamma \}\)

Where \(0 \leqslant \sum a_ix^{\gamma_i}\) if and only if \(0 \leqslant \sum_I a_ix^{\gamma_i}\) where \(I = \{\text{min. terms}\}\)

(A similar construction appears in Bowler and Su's work on classification of stringent hyperfields)

More Generally

We have an exact sequence of multiplicative groups:

$$\begin{align}1 \to B^\times \xrightarrow{\iota^\bullet} B[\Gamma]^\times \xrightarrow{v^\bullet} \Gamma \to 1\end{align}$$

Coming from morphisms of idylls

$$\begin{align}B \xrightarrow{\iota} B[\Gamma] \xrightarrow{v} \Gamma^{\rm idyll}\end{align}$$

Such that exactness for groups: \(\im(\iota^\bullet) = \eq(v^\bullet, 1)\) extends to exactness for idylls: \(\im(\iota) = \eq(v, 1)\)


\(1 \to \K^\times \to \T^\times \to \R \to 1\)

\(1 \to \K^\times \to \Gamma \to \Gamma \to 1\)

\(1 \to \T_m^\times \to \T_{m + n}^\times \to \R^n \to 1\) (where \(\T_m = \K[\R^m] = (\R^m, \le_{\rm lex})^{\rm idyll}\))

\(1 \to \S^\times \to \TR^\times \to \R^\times \to 1\) (tropical reals)

\(1 \to \P^\times \to \TC^\times \to \R^\times \to 1\) (tropical complexes, phase idyll)


A factorization is \(0 \leqslant f(x) - (x - a)g(x)\)

The multiplicity of \(f\) at \(a\) is \(0\) if \(f\) doesn't factor

and otherwise \(\mult^B_a f = 1 + \max_g \mult^B_a g\) over all factorizations

(Definition comes from the work of Baker and Lorscheid on multiplicities over hyperfields)

Morphisms and Multiplicities

Main Results

Factorizations of initial forms can be lifted

Implies \(\mult^{B[\Gamma]}_{at^\gamma} f = \mult^B_a (\In_\gamma f)\)

Example 1

\(f = 2 + 1x + 0x^2 + 0x^3 + 2x^4 + 1x^5 \in \T[x]\)

Initial forms: \(\In_1 f = 0 + x + x^2, \In_0 f = x^2 + x^3, \In_{-\frac12} f = x^3 + x^5\)

Multiplicities: \(\mult^\T_1 f = 2, \mult^\T_0 f = 1, \mult^\T_{-\frac12} f = 2\).

Example 2

Catalan OGF is a solution to \(C = 1 + tC^2\)

Consider \(f(x) = +t^0 - t^0 x + t^1 x^2 \in \TR[x]\)

Initial forms: \(\In_0 f = 1 - x, \In_{-1} f = -x + x^2 = x(x - 1)\)

Conclusion: one positive root with valuation \(0\) and one positive root with valuation \(-1\)

$$\begin{align}C_1 = 1 + t + 2t^2 + 5t^3 + \cdots, C_2 = \frac1t - 1 - t - 2t^2 - \cdots\end{align}$$

Degree Bound

If \(\sum_b \mult^B_b f\) for all polynomials in \(B[x]\) then the same is true for any tropical extension of \(B\)

Conclusion: degree bound holds for \(\text{fields}, \K, \S\) and tropical extensions by these base idylls


Paper: Tropical Extensions and Baker-Lorscheid Multiplicities for Idylls Math.RA