In my Longman, on is in parentheses, which probably means that both (1) to brush up on my English, and (2) to brush up my English are correct English?
What do you think of the second variant (without 'on')?
I have checked several online dictionaries... none answers this question... I have also checked COCA. There are a few examples analogous to (2) there but there is no guarantee these examples are grammatical...
mus-te
Top answer
i think brush up sounds well
— Anonymous
i think brush up sounds well
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It has been my experience in AmE that these have different uses, but there is some overlap.
I'll try to explain. Brush up on = review something, refresh one's memory about something, revive, etc. Brush up = (1) make one's appearance tidy (typically hyphenated as in wash and brush-up); or (2) bring to a refined state (similar to that use of the word polish).