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.