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I want to write a sentence like
We consider the equation f(x)=0. We show (the) existence of (a) solution and (the) uniqueness of (the) solution.
I want to know if the above sentence uses correctly "the" and "a". In particular, I don't know if I should put "the" in front of "existence" or "uniqueness".
I know that "the" is specific and "a" is general. But I cannot decide from this which or no article is good for the above sentence.
The articles are determiners. In your sentence, a determiner is necessary before each of the nouns. This is because each of them is singular and countable, and none of them is a proper noun. This includes even uniqueness, although it would require significant and complicated context to support its countability.
Although your question does not ask about the correctness of the sentence itself, it is incomplete, because the conjunction that introduces a clause without a predicate. If you delete that, the sentence is correct:
We show the existence of a solution and the uniqueness of the solution.
Questions regarding articles in English are among the most frequently asked here at ELL, and there are many interesting discussions of the topic elsewhere. You may wish to follow this link for further reading.
We show (the) existence of (a) solution and (the) uniqueness of (the) solution.
the existence - the because there are only two options, existence or nonexistence. So it is definite.
a solution - I presume there are many "solutions".
the uniqueness - "unique" by definition is singular, so it is definite too.
the solution - Your solution (I assume there was only one).
