polynomial transformation

Let

:$P(x)\; =\; a\_0x^n\; +\; a\_1\; x^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n\}$

be a polynomial, and

:$\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n$

be its complex roots (not necessarily distinct).

For any constant {{math|c}}, the polynomial whose roots are

:$\backslash alpha\_1+c,\; \backslash ldots,\; \backslash alpha\_n+c$

is

:$Q(y)\; =\; P(y-c)=\; a\_0(y-c)^n\; +\; a\_1\; (y-c)^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n\}.$

If the coefficients of {{math|P}} are integers and the constant $c=\backslash frac\{p\}\{q\}$ is a rational number, the coefficients of {{math|Q}} may be not integers, but the polynomial {{math|cⁿ Q}} has integer coefficients and has the same roots as {{math|Q}}.

A special case is when $c=\backslash frac\{a\_1\}\{na\_0\}.$ The resulting polynomial {{math|Q}} does not have any term in {{math|y^{n − 1}}}.

Let

:$P(x)\; =\; a\_0x^n\; +\; a\_1\; x^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n\}$

be a polynomial. The polynomial whose roots are the reciprocals of the roots of {{math|P}} as roots is its reciprocal polynomial

:$Q(y)=\; y^nP\backslash left(\backslash frac\{1\}\{y\}\backslash right)=\; a\_ny^n\; +\; a\_\{n-1\}\; y^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{0\}.$

Let

:$P(x)\; =\; a\_0x^n\; +\; a\_1\; x^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n\}$

be a polynomial, and {{math|c}} be a non-zero constant. A polynomial whose roots are the product by {{math|c}} of the roots of {{math|P}} is

:$Q(y)=c^nP\backslash left(\backslash frac\{y\}\{c\}\; \backslash right)\; =\; a\_0y^n\; +\; a\_1\; cy^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n\}c^n.$

The factor {{math|cⁿ}} appears here because, if {{math|c}} and the coefficients of {{math|P}} are integers or belong to some integral domain, the same is true for the coefficients of {{math|Q}}.

In the special case where $c=a\_0$, all coefficients of {{math|Q}} are multiple of {{math|c}}, and $\backslash frac\{Q\}\{c\}$ is a monic polynomial, whose coefficients belong to any integral domain containing {{math|c}} and the coefficients of {{math|P}}. This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers.

Combining this with a translation of the roots by $\backslash frac\{a\_1\}\{na\_0\}$, allows to reduce any question on the roots of a polynomial, such as root-finding, to a similar question on a simpler polynomial, which is monic and does not have a term of degree {{math|n − 1}}. For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.

All preceding examples are polynomial transformations by a rational function, also called Tschirnhaus transformations. Let

:$f(x)=\backslash frac\{g(x)\}\{h(x)\}$

be a rational function, where {{math|g}} and {{math|h}} are coprime polynomials. The polynomial transformation of a polynomial {{math|P}} by {{math|f}} is the polynomial {{math|Q}} (defined up to the product by a non-zero constant) whose roots are the images by {{math|f}} of the roots of {{math|P}}.

Such a polynomial transformation may be computed as a resultant. In fact, the roots of the desired polynomial {{math|Q}} are exactly the complex numbers {{math|y}} such that there is a complex number {{math|x}} such that one has simultaneously (if the coefficients of {{math|P, g}} and {{math|h}} are not real or complex numbers, "complex number" has to be replaced by "element of an algebraically closed field containing the coefficients of the input polynomials")

:$\backslash begin\{align\}$

P(x)&=0\\

y\,h(x)-g(x)&=0\,.

\end{align}

This is exactly the defining property of the resultant

:$\backslash operatorname\{Res\}\_x(y\backslash ,h(x)-g(x),P(x)).$

This is generally difficult to compute by hand. However, as most computer algebra systems have a built-in function to compute resultants, it is straightforward to compute it with a computer.

If the polynomial {{math|P}} is irreducible, then either the resulting polynomial {{math|Q}} is irreducible, or it is a power of an irreducible polynomial. Let $\backslash alpha$ be a root of {{math|P}} and consider {{math|L}}, the field extension generated by $\backslash alpha$. The former case means that $f(\backslash alpha)$ is a primitive element of {{math|L}}, which has {{math|Q}} as minimal polynomial. In the latter case, $f(\backslash alpha)$ belongs to a subfield of {{math|L}} and its minimal polynomial is the irreducible polynomial that has {{math|Q}} as power.

Polynomial transformations have been applied to the simplification of polynomial equations for solution, where possible, by radicals. Descartes introduced the transformation of a polynomial of degree {{math|d}} which eliminates the term of degree {{math|d − 1}} by a translation of the roots. Such a polynomial is termed depressed. This already suffices to solve the quadratic by square roots. In the case of the cubic, Tschirnhaus transformations replace the variable by a quadratic function, thereby making it possible to eliminate two terms, and so can be used to eliminate the linear term in a depressed cubic to achieve the solution of the cubic by a combination of square and cube roots. The Bring–Jerrard transformation, which is quartic in the variable, brings a quintic into Bring-Jerrard normal form with terms of degree 5,1, and 0.

• Tschirnhaus transformation
• Adamchik transformation

• {{cite journal | last1=Adamchik | first1=Victor S. | last2=Jeffrey | first2=David J. | title=Polynomial transformations of Tschirnhaus, Bring and Jerrard | zbl=1055.65063 | journal=SIGSAM Bull. | volume=37 | number=3 | pages=90–94 | year=2003 | url=http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf | url-status=dead | archiveurl=https://web.archive.org/web/20090226035637/http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf | archivedate=2009-02-26 }}

Category:Algebra