# 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).