to file / refine
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to file
To work on a surface or workpiece made of metal, wood, or another hard material with a file in order to smooth it, shape it, or remove material.
Der Schlosser feilte die Kanten des Metallstücks glatt.
The locksmith filed the edges of the metal piece smooth.
The two meanings can be distinguished well by the grammar. The literal meaning has a direct accusative object (e.g., 'das Metall feilen' — to file the metal), while the figurative meaning always uses the preposition 'an' with the dative (e.g., 'an der Rede feilen' — to refine the speech).
Er feilt das Holzstück für sein Modellboot.
He is filing the piece of wood for his model boat.
When 'feilen' is used in the figurative sense of 'to improve' or 'to perfect', it is always used with the preposition 'an' followed by a dative object.
Der Redner hat lange an seinem Vortrag gefeilt.
The speaker spent a long time refining his talk.
In the literal meaning, 'feilen' (to file) is often used with hard materials such as 'Metall' (metal) or 'Holz' (wood). The phrase 'sich die Fingernägel feilen' (to file one's fingernails) is also very common in the cosmetic context.
Vor dem Lackieren feilt sie sich immer die Nägel.
Before applying nail polish, she always files her nails.
In the figurative meaning, abstract things are 'gefeilt' (refined). Typical objects are 'ein Plan' (a plan), 'eine Rede' (a speech), 'ein Text' (a text), 'eine Technik' (a technique), or 'eine Fähigkeit' (a skill).
Der Musiker feilt seit Wochen an seiner Technik.
The musician has been refining his technique for weeks.
to improve
'verbessern' and 'feilen an' both mean that something is meant to become better. 'verbessern' is more general: you can improve a text, performance, situation, or product. 'feilen an' sounds more precise and often more intensive. You use it when you are working on details and gradually bringing something into a better form, for example a speech, a plan, or a skill. 'verbessern' works for both large and small changes, while 'feilen an' is used more for careful work on finer details.
Vor der Bewerbung feilt Lena noch an ihrer Präsentation.
Before applying for the job, Lena continues refining her presentation.
Am Montag verbessert Ben zu Hause die Tonqualität seines Videos.
On Monday, Ben improves the sound quality of his video at home.