2024/10/22 by Lilybelle Cowland Kellock, Kellock, Lilybelle Cowland
Computer Science · Engineering · #14G20 #14H45 #14Q05 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary: 11G20 #Secondary: 14D10
paper · pdf · doi:10.48550/arxiv.2410.17148
openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the minimal regular model of an elliptic curve E/Kunr is uniquely determined by the valuation of jE and ΔE.