Results for resolution of equations in polynomial ring Is there any reference for resolution of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!
 A: In general this is a difficult problem, but in special cases like $x^n+y^n=z^n$ in polynomials we can use Mason's theorem, which is about an analogue of the $abc$-conjecture for polynomials in $\mathbb{C}[t]$. It implies the following result:
Theorem: Let $n ≥ 2$ be an integer, and suppose $a, b, c \in \mathbb{C}[t]$ are pairwise relatively prime polynomials, at least one of which is not a constant, satisfying $a^n + b^n = c^n$. Then $n = 2$.
For the case $n=2$ we have the basic solutions $(a(t),b(t),c(t))=(m(t)^2-n(t)^2,2m(t)n(t),m(t)^2+n(t)^2)$ with polynomials $m(t),n(t)$.
A: The question is vague about what equations are to be considered. As Qiaochu points out in his comment, in the most general possible interpretation, the problem is undecidable. As Robert points out in his comment, if the equation defines a curve over $\mathbb{C}$ then it can be decided a simple genus calculation. For an arbitrary curve over $\mathbb{C}(t)$ there are effective versions of the Mordell conjecture. This was proved originally by Manin and Grauert and effectively by Szpiro, with further improvements by several others (e.g. Vojta, Compositio 78 (1991), 29-36).
