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We give two algorithms to compute linear determinantal representations ofsmooth plane curves of any degree over any field. As particular examples, weexplicitly give representatives of all equivalence classes of lineardeterminantal representations of two special quartics over the field$\mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic.This paper is a summary of third author's talk at the JSIAM JANT workshop onalgorithmic number theory in March 2018. Details will appear elsewhere.

On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree