Divisors and Multiplicities under tropical and signed shadows

Sera Gunn

Part I:

Divisors on Metric Graphs

Divisors, line bundles, linear systems, etc. are connected to projective embeddings.

E.g. $\P^1 \xrightarrow{\O(3)} \P^3$ \[[x:y] \mapsto [x^3 : x^2y : xy^2 : y^3]\]

Twisted Cubic

Divisors, line bundles, linear systems, etc. are connected to projective embeddings.

E.g. $\P^1 \xrightarrow{\O(3)} \P^3$ \[[x:y] \mapsto [x^3 : x^2y : xy^2 : y^3]\]

Twisted Cubic

Can embed graphs in $\R^3$
putting vertices on the TC

$p^3 + px - 2py - 3px^2 + 2xy + py^2 - 2p^3x^3 + px^2y + pxy^2 - p^3y^3 \in \Q_p[x,y]$

Degree 3 Newton Polygon Tropical Elliptic Curve

Newton polytope/complex
and associated tropical curve

Degree 3 Newton Polygon Tropical Elliptic Curve

The metric graph associated to a tropicalization is called a skeleton

Lattice Length Metric

Lattice length metric

Tropical Balancing Condition

Tropical balancing condition \[ \sum \omega_e \vec{v}_e = \vec0 \]

Degree 3 Newton Polygon minimal-skeleton

Can also contract leaf vertices to obtain other skeleta

Going backwards

minimal-skeleton tropical hexagon

Going backwards

tropical hexagon tropical hexagon with rays

Baker-Rabinoff

There exist coordinate functions, which lift, such that the resulting tropicalization is an isometry (tropical weights are $1$) on a fixed skeleton.

Baker-Rabinoff coordinate functions

Baker-Rabinoff

Fixed skeleton—rays have positive weights

Baker-Rabinoff coordinate functions

G-Jell

There exist coordinate functions, which lift, such that the resulting tropicalization is an isometry on the extended skeleton (considering all the rays).

Semistable Model

Components of $\X_{\F_p}$ give valuations on $\X$

Semistable Model

Dual graph of $\X_{\F_p}$ is a skeleton

If $f$ is a meromorphic function on $\X^{\mathrm{an}}$ then \[ (x, |\cdot|) \mapsto \log |f(x)| \] is a piecewise linear function.

Part II:

Multiplicities

Tropical multiplicities

Every univariate cycle is a sum of degree 1 cycles

Newton, Baker-Lorscheid, G

# of roots with valuation $\gamma$ equals
the width of the edge of slope $-\gamma$

Enriched Multiplicities

Sign-enriched univariate cycle

G (Combining Newton and Descartes)

# of pos. roots with valuation $\gamma$ equals
the number of sign changes in the edge of slope $-\gamma$

Descartes' Rule

Quintic: y = 1 - 3x - 2x^2 + x^5

\[ y = 1 - 3x - 2x^2 + x^5 \]

\[ +, -, -, \textcolor{#CCC}0, \textcolor{#CCC}0, + \]

Multivariate Descartes Problem

  • Given a system of equations $f_1 = \dots = f_n = 0$,
  • with just information about the signs of the coefficients,
  • what can we say about the signs of $V(f_1,\dots,f_n)$?
  • Lower bound on # of positive roots by Itenberg-Roy using patchworking (combinatorial)
  • Their lower bound is not sharp (example by Li-Wang)
Goal: algebraic mulitiplicity that explains these

Arithmetic in the Shadows

\[ y = 1 - 3x - 2x^2 + x^5 \]

\[ +, -, -, \textcolor{#CCC}0, \textcolor{#CCC}0, + \]

\[ \begin{aligned} (1 - x)(1 &- x - x^2 - x^3 - x^4) \\ = 1 &- x - x^2 - x^3 - x^4 \\ &- x + x^2 + x^3 + x^4 + x^5 \\ \ni 1 &-x - x^2 \hspace{1.54cm} +x^5 \end {aligned} \]

Baker-Lorscheid Multiplicities

Sign-enriched univariate cycle

Descartes

Number of sign changes (combinatorial)

Baker-Lorscheid

Maximum number of times one can factor out $(x - 1)$ using sign arithmetic

Baker-Lorscheid Multiplicities

Every univariate cycle is a sum of degree 1 cycles

Newton

Width of the edge with slope $-\gamma$ (combinatorial)

Baker-Lorscheid

Maximum number of times one can factor out $(x + \gamma)$ using tropical arithmetic

Tropical Hyperfield (min/plus arithmetic)

\[\T = \R \cup \{\infty\}\] \[ x \cdot_\T y = x +_\R y \quad \text{(mult. is add.)} \] \[ x_0 \in x_1 \boxplus \dots \boxplus x_n \iff \min\{x_0,\dots,x_n\} \text{ occurs twice} \]

Geometric Multiplicities

Addition of tropical cycles

Number of copies of a line in a tropical cycle

(Can be adapted to include signs)

If a subdivision isn't given, we can impose one

Perturbation multiplicity example

Boundary Multiplicity

Boundary multiplicity example

\[ \bmult = \min_i \mult|_{x_i = 0} \]

Systems of Equations

(Putting it all together)

Li-Wang

$a, b, r, s, t > 0$

\[ \begin{cases} f\coloneqq 1+ax-by = 0\\ g\coloneqq 1+rx^3-sy^3-tx^3y^3=0 \end{cases} \]

Has $3$ positive roots for some choice of coefficients

Itenberg-Roy predict $2$

\[ \begin{cases} f\coloneqq 1+ax-by = 0\\ g\coloneqq 1+rx^3-sy^3-tx^3y^3=0 \end{cases} \]

Mixed sparse resultant: \[ R \propto \prod_{p \in V(f,g)} (p_1 u + p_2 v + p_3 w) \]


					ring R = (0,(u,v,a,b,r,s,t)),(x,y),dp;
					ideal I = 1+ux+vy, 1+ax-by, 1+rx3-sy3-tx3y3;
					det(mpresmat(I,0));
				

\[ \begin{cases} f\coloneqq 1+ax-by = 0\\ g\coloneqq 1+rx^3-sy^3-tx^3y^3=0 \end{cases} \]

Mixed sparse resultant: \[ R \propto \prod_{p \in V(f,g)} (p_1 u + p_2 v + p_3 w) \]

Signed resultant in LW example
Signed resultant in LW example

\[\downarrow\]

Signed resultant in LW example
Signed resultant in LW example

\[ \bmult R \le 3 \]

Other Highlights

G

For univariate polys, $\mult f = \mult(\In f)$

G

$\sum \mult f \le \deg f$ for stringent hyperfields
meaning: $a \boxplus b = \{c\}$ if $0 \notin a \boxplus b$

Gross-G

New proof of Itenberg-Roy's lower bound using resultants