In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.

Simple examples

Translating the roots
Let 

be a polynomial, and 
 
be its complex roots (not necessarily distinct).

For any constant , the polynomial whose roots are 
 
is 

If the coefficients of  are integers and the constant  is a rational number, the coefficients of  may be not integers, but the polynomial  has integer coefficients and has the same roots as .

A special case is when  The resulting polynomial  does not have any term in .

Reciprocals of the roots
Let 

be a polynomial. The polynomial whose roots are the reciprocals of the roots of  as roots is its reciprocal polynomial

Scaling the roots

Let 

be a polynomial, and  be a non-zero constant. A polynomial whose roots are the product by  of the roots of  is 

The factor  appears here because, if  and the coefficients of  are integers or belong to some integral domain, the same is true for the coefficients of .

In the special case where , all coefficients of  are multiple of , and  is a monic polynomial, whose coefficients belong to any integral domain containing  and the coefficients of . 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 , 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 . For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.

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

be a rational function, where  and  are coprime polynomials. The polynomial transformation of a polynomial  by  is the polynomial  (defined up to the product by a non-zero constant) whose roots are the images by  of the roots of .

Such a polynomial transformation may be computed as a resultant. In fact, the roots of the desired polynomial  are exactly the complex numbers  such that there is a complex number  such that one has simultaneously (if the coefficients of  and  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")

This is exactly the defining property of the resultant

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.

Properties
If the polynomial  is irreducible, then either the resulting polynomial  is irreducible, or it is a power of an irreducible polynomial. Let  be a root of  and consider , the field extension generated by . The former case means that  is a primitive element of , which has  as minimal polynomial. In the latter case,  belongs to a subfield of  and its minimal polynomial is the irreducible polynomial that has  as power.

Transformation for equation-solving

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  which eliminates the term of degree  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.

References
 

Algebra