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

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 $\Delta_E$.