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Minimization and reduction of plane curves

Michael Stoll

When given a plane curve over $$\mathbf Q$$, it is usually desirable (for computational purposes, for example) to have an equation for it with integral coefficients that is ‘small’ in a suitable sense. There are two aspects to this. One is to find an integral model with invariants that are as small as possible, or equivalently, a model that has the best possible reduction properties modulo primes. This is called minimization. The other, called reduction, is to find a unimodular transformation of the coordinates that makes the coefficients small. I will present a new algorithm that performs minimization for plane curves of any degree (this is joint work with Stephan Elsenhans) and also explain how one can perform reduction (based on some older work of mine).