"at all the vertices", what does this mean?

Solution 1:

It means "[The] function is non-negative at all of the vertices in the structure S and positive at some vertex". The of isn't necessary though it may help to clarify the meaning.

There may be some confusion between the phrases not negative at all and non-negative at all. Some examples may help to clarify what I mean. Note that these sentences do not have the same meaning as your original sentence.

This sentence asserts that the function is never negative.

The function is not negative at all.

It could also be written as

The function is not at all negative.

The phrase at all serves to emphasize not.

However in your sentence the phrase at all refers to the set of vertices. It could be written as

The function is not negative at all the vertices ...

(Though, as jwpat points out, there are better ways to say this.)

If it were reordered it would not make sense. Thus at all is not serving to emphasize not.

*The function is not at all negative the vertices ...

This question discusses the difference between at all and not at all.

Solution 2:

[The] function is non-negative at all the vertices of the structure S and positive at some vertex...

That statement has a specific meaning, and it is a proper and precise way of stating that meaning:

The function is not negative at any vertex of S (that is, there is no vertex of S where the function is negative). In addition, there exists at least one vertex of S where the function is positive.

Note that zero is neither negative nor positive; to say that a function is non-negative means that its value is zero or positive. To say a function is positive means that it has a value greater than zero.

Most native English speakers will not notice any difference due to of being present or absent in the example sentence. If you are concerned about it, change all to every:

The function is non-negative at every vertex of the structure S and is positive at some vertex...

Edit: onomatomaniak commented,

I wonder, though, if “some vertex” means not at least one vertex, as you indicate, but rather [precisely] one vertex. If I wanted to indicate at least one, my inclination would be to write some vertices, not some vertex.

Because phrasing like “positive at one vertex of S and zero at all others” is the obvious way to express a single-vertex-positive condition, I think it would be perverse or misguided for someone to write “non-negative at all the vertices of S and positive at some vertex” to mean positive at one and only one vertex.

Anyhow, the usual intent of phrasing like “non-negative at all the vertices of the structure S and positive at some vertex” is to describe a function that never is negative, and by dint of going positive somewhere, is non-trivial.

Regarding “some vertices” vs “some vertex”: (1) In mathematical writing one desires to make premises only as strong as necessary for a proof to go through. If we take “some vertices” as implying or suggesting multiple vertices, and “some vertex” as one or more, then the latter condition is weaker, hence desired. (2) In a proof, “some vertex” is likely to be part of a phrase like “some vertex, say v...” – a construction where “some vertices” would not work.

Solution 3:

"At all" has its special meaning only in a phrase with a Negative Polarity trigger such as no, none, never, or a yes-no question. Words with the prefix "non-", despite their negative meaning, are not grammatically Negative Polarity words and so the phrase at all cannot occur in this sentence and that particular parse is impossible.