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

TL;DR

The work presents a constructive method to recover the cluster picture of a degree-$d$ polynomial over a discretely valued field from valuations of coefficient polynomials, via a finite explicit family of polynomials $g_d^{(i)}$ tied to rational functions $J_G$ associated with auxiliary graphs. By introducing the averaging function $A_{n+1}$ and an inductive graph-based framework, it shows how the full root-distance configuration can be recovered without solving for the roots themselves, and it derives hyperelliptic-curve consequences such as the dual graph of the special fibre, inertia action on the Jacobian, and the conductor exponent; this mirrors Tate-type information for elliptic curves in the high-char-avoidance setting. A degree-5 algorithm is given with explicit graphs and valuations, and an implementation demonstrates practical efficiency gains over existing methods. The results provide a concrete, coefficient-only pathway to determine reduction-type data for hyperelliptic curves from their defining polynomials.

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 $\{(\mathrm{ord}(x_i-x_j),i,j) : 1\leq i< j \leq d \}$ where $x_1,\dots,x_d$ are the roots of $f(x)$, can be recovered without knowing the roots of $f(x)$ over $\bar{K}$. We construct an explicit list of polynomials $g_d^{(1)},\dots,g_d^{(t_d)}\in\mathbb{Z}[A_0,\dots,A_{d-1}]$ such that the valuations $\mathrm{ord}(g_{d}^{(i)}(a_0,\dots,a_{d-1}))$ for $i=1,\dots,t_d$ uniquely determine this set of distances for the polynomial $f(x)=c_f(x^d+a_{d-1}x^{d-1}+\dots+a_0)$, and we describe the process by which they do so. We use this to deduce that if $C:y^2=f(x)$ 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 $C/K^{\mathrm{unr}}$, 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 $p\geq 5$ the dual graph of special fibre of the the minimal regular model of an elliptic curve $E/K^{\mathrm{unr}}$ is uniquely determined by the valuation of $j_E$ and $Δ_E$.

Figures (11)

• Figure :
• Figure : $G_{0}(f)$
• Figure : $G_{1}(f)$
• Figure : $G_{2}(f)$
• Figure : $G_{3}(f)$

Theorems & Definitions (52)

• Theorem 1.1: =Theorem \ref{['polythm']}
• Theorem 1.2: See Theorem \ref{['firstalgorithm']}
• Definition 1.3
• Example 1.4
• Definition 1.5
• Example 1.6
• Remark 1.7
• Theorem 1.8: =Theorem \ref{['firstalgorithm']}
• Definition 1.9: See Definition \ref{['dnf']}
• Definition 1.10: See Definitions \ref{['aux']} and \ref{['curlyg']}