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So this is a homework question in my algebra class that I'm getting really stuck on... it should be straightforward, but I'm not sure how to interpret the differential equation. Any hints (solutions are nice too, but I really want to solve this) would be greatly appreciated!

Let $f(x)=a_0x^n+...+a_n$.

a) (I can do this part) Show that $f(x)$ and $f(x+c)$ have the same discriminant for all $c$.

b) (Part I'm stuck on). Let $D=\text{Disc}(f) \in \mathbb{Z}[a_0, ..., a_n]$. Use a) to show that $D$ satisfies the following differential equation holds:

$na_0\frac{\partial D}{\partial a_1}+(n-1)a_1\frac{\partial D}{\partial a_2} + \cdots + a_{n-1}\frac{\partial D}{\partial a_n} = 0$.

user193763
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2 Answers2

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I would suggest computing $f(x+c) -f(x) =0$ as a Taylor series, so that the first coefficient, linear in $c$ will contain only first derivatives, and vanish. You might want to use the definition of the discriminant in terms of roots, or otherwise, also. Hope it helps.

Rogelio Molina
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$0=\frac{\partial D(a_0,\ldots,a_n)}{\partial c}=\frac{\partial D(a_0,na_0c+a_1,n(n-1)/2a_0c^2+(n-1)a_1c+a_2\ldots,a_0c^n+\dots+a_n)}{\partial c}$.

Then use chain rule.

Wei Xia
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