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}

Idylls

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

a monoid-with-zero \(B^\bullet = (B^\bullet, 0_B, 1_B, \cdot_B)\)
a proper ideal \(N_B \subset \N[B^\bullet]\)

Such that \(B\) is field-like

\(0_B \neq 1_B\)
\(B^\bullet\) is a group-with-zero
there exists a unique \(-1\) with \((-1)^2 = 1 \text{ and } 1 + (-1) \in N_B\)

Here

\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 \(\N[B^\bullet]\):

Reflexive: \(x \leqslant x\)
Transitive: \(x \leqslant y \wedge y \leqslant z \implies x \leqslant z\)
Additive and multiplicative: \(u \leqslant v \wedge x \leqslant y \implies (u + x \leqslant v + y) \wedge (ux \leqslant vy)\)
Antisymmetric on \(B^\bullet\) (not 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

Examples

\(\F_{1^n}\) 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 \(B\)

Free \(B\)-algebra
\(\sum b_ix^{n_i} \leqslant \sum c_ix^{m_i}\) if and only if the relation holds in each degree

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: \(\im(\iota) = \eq(v, 1)\)

Examples

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

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

\(1 \to \T_m^\times \to \T_{m + n}^\times \to \R^n \to 1\) (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)

Multiplicities

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

Morphisms preserve factorizations
If \((x - a)\) can be factored out \(k\)-times then the same is true after morphism
Therefore \(\mult^B_a f \le \mult^C_{\varphi(a)} \varphi(f)\) for any morphism \(\varphi : B \to C\)
Also, initial forms are morphisms so \(\mult^{B[\Gamma]}_{at^\gamma} f \le \mult^B_a (\In_\gamma f)\)

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

End

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