Tropical Extensions and Baker-Lorscheid Multiplicities

Trevor Gunn

November, 2022


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