I'm trying to figure out a set of solutions to an ideal membership problem. Namely, if $f(x,y),g(x,y)\in \mathbb{C}[x,y]$ I want to figure out what constraints I need such that they satisfy $$g(x,y)-2xf(x,y)\in (y-x^2).$$ Since we're in a UFD, and commutative, this is the same as finding a set of solutions $f,g,h \in \mathbb{C}[x,y]$ to the following equation: $$g(x,y)-2xf(x,y) = h(x,y)(y-x^2).$$ This looks like a diophantine equation over $\mathbb{C}[x,y]$ to me and I was wondering if anyone knew any methods to solve problems like this. Thanks.
2026-03-27 02:50:51.1774579851
Diophantine Equations over polynomial rings
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