I am reading a cubic formula derivation here:
http://danielrui.com/papers/cubicPolynomial.pdf
It looks fairly straight forward. The author defined:
$y = \sqrt[3]{u} − \sqrt[3]v$
so far so good, I suppose there can always be a $u$ and $v$ that will work. But then, he proceeds with
"Let's define $v − u = e$ ('cause we can)"
I am lost here. $e$ is not just any constant. It is
$e = -\frac{a^3}{27}+\frac{a^3}{9}-\frac{ab}{3}+c$
How can he make that claim?
When you make $y=\sqrt[3]{u}-\sqrt[3]{v}$ you are simply renaming your variables. Nothing prevents you from assuming that this two new variables also satisfy another relation between them (think of this as if you go from one single equation to a system of equations). In this particular case, the most clever relation to impose between $u$ and $v$ is $v-u=e$. Note that $e$ does not depend on $y$, as a comment points out.
Oops that's my old article I wrote in high school! You're right there are some subtle gaps I left; in fact after I wrote it I ended up having more questions about it (see Cubic Formula Derivation Check and Questions) and to this day I'm still not 100% sure how everything works.
Regarding your question, here's how I would explain it now: our goal is to show that there exist some (perhaps complex...) numbers $u,v$ such that $u-v=-e$ and $\sqrt[3]u-\sqrt[3]v$ is a root of the cubic $y^3+dy+e=0$.
The key idea (which appears everywhere in math) is that to prove some things exist, one can first ASSUME they exist, make some logical deductions --- i.e. prove some properties that the things must satisfy (if they existed), or show that the solutions must be of some given form/formula --- and then CHECK that the forms/formulas you found actually are the things you originally were looking for.
So I can rephrase the logic of my article as follows:
ASSUME that there are numbers $u,v$ s.t. (1) $u-v=-e$ and (2) $\sqrt[3]u-\sqrt[3]v$ is a root of the cubic $y^3+dy+e=0$ (these I refer to as the "2 initial assumptions"). THEN, $u,v$ have to additionally satisfy $uv = d^3/27$ and $$u=\frac{-e \pm \sqrt{e^2+4d^3/27}}2 \quad \text{ and } \quad v=\frac{e \pm \sqrt{e^2+4d^3/27}}2.$$ Then one can check that these specific values I've written for $u,v$ actually do satisfy the 2 initial assumptions.
I think it is possible to do this last "checking" step by actually plugging these massive formulas in for $y$ and then cubing $y$ and so forth; but I argue that I actually have done the work: I can rephrase the logic of my article one more:
Suppose we have numbers $u,v$ s.t. $u-v=-e$ (so at this point $u,v$ are specific fixed numbers). Then, the 2nd page of my article essentially shows that: $\sqrt[3]u-\sqrt[3]v$ (a specific number based on the previous fixed values of $u,v$) is a root of the cubic $y^3+dy+e=0$ if and only if $uv=d^3/27$.
Thus in order to shows that the massive formulas above --- which I'll call $u_0, v_0$ respectively --- satisfy the 2 initial assumptions, the "if and only if" above tells us that it suffices to check that $u_0v_0 = d^3/27$, which is quite easy to do.
To summarize, I ASSUMED that $u,v$ existed. Then I derived formulas for them. Then I checked that these formulas worked, and so indeed $u,v$ DO exist.
P.S. in the years since I wrote my article, I think the best resource for learning the cubic formula is this Mathologer video: https://www.youtube.com/watch?v=N-KXStupwsc&ab_channel=Mathologer. He actually gives geometric/visual intuition for the algebraic substitutions I made in my article.
