Is my Major Professor Counted as a Co-Author?
Using &quot;we&quot; as you suggest is fine.  As to the APA, I say a mathematician needs a guide to psychology writing like a fish needs a bicycle.  We with a single author is good enough for Terry Tao 1,2 so you can do it as well.
There are many specific style issues in mathematics.  There are many guides. Here is one 3.
By the way, the best time to discuss authorship issues with your professor is back when you started this project.  The second best time is today.