Using "we have" in maths papers
Solution 1:
In my opinion it is even good style. You are involving the reader somehow to the discussion, if you write phrases like "we have", "we consider", "we may assume", "one can see that" etc.
By the way, it is common in other languages as well and you can read similar phrases in papers of famous mathematicans all over the world.
Some examples:
Noi sappiamo (...) $\tag{Fubini, 1903}$ (we know that, ... Ref, p.6 )
or
(..) nehmen wir an, dass (..) $\tag{Minkowski, 1900}$ (We assume that ... Ref)
or
However in the reducible case (..) we have to consider (..) $\tag{Wiles, 1999}$ (Ref, p.4)
Furthermore I found a guide by MIT in which is said
Be forthright: write in an unhesitating, straightforward, and friendly style, ridding your language of needless and bewildering formality. Be wary of awkward and inefficient passive constructions. Often the passive voice is used simply to avo id the first person. However, the pronoun “we” is now generally considered acceptable in contexts where it means the author and reader together, or less often, the author with the reader looking on.
Solution 2:
It is not too colloquial -- it is used all over the place in papers. In your example, "we have" is preferable (IMO) to "there exists the," which is verbose and a bit ugly.
Solution 3:
(EDIT:) The basic difference is that 'we' suggests two minds, specifically a teacher/student paradigm.
This is perfect for any teaching context.
But academic peers exchanging information will go to absurd lengths to avoid insulting their colleagues by insinuating such a relationship.
The object of maths is understanding.
The object of mathematical writing is communicating that understanding with the utmost clarity.
And at the heart of communication is dialogue. It is one person explaining to another person.
Look at the Greek dialogues (e.g. http://www.sacred-texts.com/cla/plato/laches.htm), and observe how easy it is to understand the material when both parties are voiced, even if the student's part is little more than a device.
This duality is critical; question and answer complement one another; the question creates a space into which the answer may manifest, the answer provokes further questions, and so on. This creates an interconnected web of understanding.
And if the 'we' is lost, then the dialogue is also lost.
Look at Einstein's famous papers (e.g. http://milesmathis.com/five.html) and notice how you feel he is looking over your shoulder and explaining something to you.
It is unfortunate that academic papers feel the need to be absolutely impersonal, because it makes them dry. I have to read through a lot of them, and the conventional style is a hindrance; you should break every rule in the pursuit of clarity.
Interestingly, as YouTube videos are gaining popularity over written text as teaching resources, we are seeing a resurgence of the Socratic Dialogue paradigm! As a video is typically made in real-time, the presenter doesn't have the luxury of formalising the presentation. This is a good thing!
Solution 4:
"We have" is just an expression that is meant to involve the reader. It's a little less dry than "there exists"; it's a bit of a colloquialism, and a little friendlier than the passive voice or third person. I think it's a good stylistic choice.
Here's an example to illustrate the point, from a different field entirely:
"Looking at the top-right corner of Van Gogh's Starry Night, there is a bright-yellow moon, a great contrast that draws one's attention."
vs.
"Looking at the top-right corner of Van Gogh's Starry Night, we can see a bright-yellow moon, a great contrast that draws our attention."
I think the latter sounds more pleasant than the former, because it is less stilted and less distant. The case for wanting to use "we can see that" or "we have that", etc. in mathematical writing is similar.
For reasons similar to the above, I sometimes even use expressions like "then our sequence converges" instead of "then the sequence converges", which is perhaps a little more controversial.