My question might be an easy or could be a bit complicate and classic. Actually I am trying to understand why the discriminant of a binary quadratic form is a "the fundamental invariant" under $GL(2,\mathbb{Z})$action i.e any other invariant is a polynomial of the discriminant. Also I am interested to know about general binary forms. More precisely I would like to know the number of fundamental invariant of binary form of degree n under $GL(2,\mathbb{Z})$action.
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Finding the number of generators for the invariants of binary forms is a classic and very hard problem in invariant theory. For forms of small degree one can find a description in Hilbert's book on invariant theory (ISBN 9780521449038). In the 19th century it was solved for forms of degree at most 8, and it has recently been pushed to degree 10 in http://dx.doi.org/10.1016/j.jsc.2010.03.002 using computer calculations (where one needs 106 generators). The latter link gives the history of the problem in more detail. 

