Tschirnhaus Transformation yields Depressed Polynomial
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Theorem
Let $\map f x$ be a polynomial of order $n$:
- $a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$
Then the Tschirnhaus transformation:
$y = x + \dfrac {a_{n - 1} } {n a_n}$
converts $f$ into a depressed polynomial:
- $b_n y^n + b_{n- 1} y^{n - 1} + \cdots + b_1 y + b_0$
where $b_{n - 1} = 0$.
Proof
Substituting $y = x + \dfrac {a_{n - 1} } {n a_n}$ gives us:
- $x = y - \dfrac {a_{n - 1} } {n a_n}$
By the Binomial Theorem:
- $a_n x^n = a_n \paren {y^n - \dfrac {a_{n - 1} } {a_n} y^{n - 1} + \map {f'_{n - 2} } y}$
where $\map {f'_{n - 2} } y$ is a polynomial in $y$ of order $n - 2$.
Now we note that:
- $a_{n - 1} x^{n- 1} = a_{n - 1} y^{n - 1} - \map {f_{n - 2} } y$
where $\map {f_{n - 2} } y$ is another polynomial in $y$ of order $n - 2$.
The terms in $y^{n - 1}$ cancel out.
Hence the result.
$\blacksquare$
Source of Name
This entry was named for Ehrenfried Walther von Tschirnhaus.