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$\mathbb{C}(f,g)=\mathbb{C}(t)$ and $(f'(t),g'(t)) \neq 0$, but $\mathbb{C}[f,g]\subsetneq \mathbb{C}[t]$
algebraic-geometry
polynomials
commutative-algebra
field-theory
resultant
Characterizing $f$ and $g$ such that $\deg(\gcd(f,g)) \geq 2$.
polynomials
discriminant
resultant
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