# Recovering the cluster picture of a polynomial over a discretely valued field

## TL;DR

This paper shows how to determine the cluster picture of a polynomial over a discretely valued field without knowing its roots.

## Contribution

The paper introduces a method to recover the cluster picture using valuations of explicit polynomials in the coefficients.

## Key findings

- Valuations of specific polynomials in the coefficients determine the cluster picture of the polynomial.
- This method applies to hyperelliptic curves and determines properties of their minimal models.
- The approach generalizes a result from Tate's algorithm for elliptic curves to hyperelliptic curves.

## Abstract

For
f(x)
, a separable polynomial of degree
d
over a discretely valued field
K
, we describe how the cluster picture of
f(x)
over
K
, in other words, the set of tuples
{(ord(xi−xj),i,j):1≤i<j≤d}
, where
x1,…,xd
are the roots of
f(x)
, can be recovered without knowing the roots of
f(x)
over
K¯
. We construct an explicit list of polynomials
gd(1),…,gd(td)∈ℤ[A0,…,Ad−1]
such that the valuations
ord(gd(i)(a0,…,ad−1))
for
i=1,…,td
uniquely determine this set of distances for the polynomial
f(x)=cf(xd+ad−1xd−1+⋯+a0)
, and we describe the process by which they do so. We use this to deduce that if
C:y2=f(x)
is a hyperelliptic curve over a local field
K
. This list of valuations of polynomials in the coefficients of
f(x)
uniquely determines the dual graph of the special fibre of the minimal strict normal crossings model of
C/Kunr
, the inertia action on the Tate module and the conductor exponent. This provides a hyperelliptic curves analogue to a corollary of Tate’s algorithm, that in residue characteristic
p≥5
, the dual graph of special fibre of the minimal regular model of an elliptic curve
E/Kunr
is uniquely determined by the valuation of
jE
and
ΔE
.