I am getting to like prime ideals 👉🏼 en.m.wikipedia.org/wiki/Prime_ide…
Perhaps it’s more accurate to say I like all mathematical objects which reveal their secrets to me slowly…#mathematicallife
The same is not necessarily true of people …
Take an ODE like w”=6w²-g₂/2, where w is a function of t and g₂ is a given constant.
We can integrate this ODE after multiplying it by w’ (the first derivative of w).
After multiplication we have
w’w”=6w²w’-g₂w’/2
and since both sides are derivatives of something, we can integrate to get
w’w”=6w²w’-g₂w’/2
and since both sides are derivatives of something, we can integrate to get
(w’)² = 4w³-g₂w-g₃ , where g₃ is an integration constant.
Or, writing y=w’, x=w, we have a polynomial f(x,y)=0, where f(x,y):= y² - 4x³+g₂x+g₃.
This ability to view the solutions of an ODE from two perspectives (1. as functions of time; 2. as parametrizing algebraic curves) is the start of an amazingly important development in my field…
In algebraic geometry (perspective #2), this polynomial generates a prime ideal of ℂ[x,y].
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