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

For $f\left(x\right)$ , a separable polynomial of degree $d$ over a discretely valued field $K$ , we describe how the cluster picture of $f\left(x\right)$ over $K$ , in other words, the set of tuples $\left\{\right(\text{ord}(x_i-x_j),i,j):1\le i<j\le d\}$ , where $x_1,\dots ,x_d$ are the roots of $f\left(x\right)$ , can be recovered without knowing the roots of $f\left(x\right)$ over $\overline{K}$ . We construct an explicit list of polynomials $g_d^{\left(1\right)},\dots ,g_d^{\left(t_d\right)}\in \mathbb{Z}[A_0,\dots ,A_{d-1}]$ such that the valuations $\text{ord}\left(g_d^{\left(i\right)}\right(a_0,\dots ,a_{d-1}\left)\right)$ for $i=1,\dots ,t_d$ uniquely determine this set of distances for the polynomial $f\left(x\right)=c_f(x^d+a_{d-1}{x}^{d-1}+\cdots +a_0)$ , and we describe the process by which they do so. We use this to deduce that if $C:y^2=f\left(x\right)$ is a hyperelliptic curve over a local field $K$ . This list of valuations of polynomials in the coefficients of $f\left(x\right)$ uniquely determines the dual graph of the special fibre of the minimal strict normal crossings model of $C/K^{unr}$ , 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 $p\ge 5$ , the dual graph of special fibre of the minimal regular model of an elliptic curve $E/K^{unr}$ is uniquely determined by the valuation of $j_E$ and ${\Delta }_E$ .