Polynomial transformation

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

Simple examples

Translating the roots

Let

$P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}$

be a polynomial, and

$\alpha _{1},\ldots ,\alpha _{n}$

be its complex roots (not necessarily distinct).

For a any constant c, the polynomial whose roots are

$\alpha _{1}+c,\ldots ,\alpha _{n}+c$

is

$Q(y)=P(y-c)=a_{0}(y-c)^{n}+a_{1}(y-c)^{n-1}+\cdots +a_{n}.$

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

Reciprocals of the roots

Let

$P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}$

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

$Q(y)=y^{n}P\left({\frac {1}{y}}\right)=a_{n}y^{n}+a_{n-1}y^{n-1}+\cdots +a_{0}.$

Scaling the roots

Suppose we have some polynomial $a_{0}+a_{1}x+\cdots +a_{n-1}x^{(n-1)}$ and we want to find some polynomial that has roots that are k units greater than the roots of f(x). One obvious approach would be to find the roots of f(x) and then multiply out a polynomial that has k greater than those roots. Also notice that this can be easily accomplished by thinking of the graph of the function. If it has roots at $r_{1},r_{2}\ldots r_{n}$ than we can simply shift the function over k units which will give us a function that has roots that are k units greater than the roots of f(x). so for x in f(x) substitute x − k.

General method

Suppose we have a polynomial f(x) of degree d, and we want to find the polynomial g(x), whose roots are the d values (possibly not all distinct) of a given polynomial m(x) of degree e evaluated at the roots of f. The method is based on the fact that the coefficients of g are symmetric functions of its roots, and thus symmetric functions of the roots of f. It follows that each coefficient of g is a polynomial in the elementary symmetric polynomials of the roots of f, which are, up to their sign, the coefficients of f.

Using above remark without care implies huge computations with multivariate polynomial, which are impossible, in practice, even for small degrees of f and m. This may be avoided by using power sums and Newton's identities.

Firstly one computes the d first power sums of m up to m^d and expands them to get:

$m^{k}(x)=\sum _{i=1}^{ek}\alpha _{i,k}x^{i}.$

Thus the sum of the k powers of the roots of the desired polynomial g is

$\sum _{\alpha \;{\text{root of}}\;f}m^{k}(x)=\sum _{i=1}^{ek}\alpha _{i,k}\left(\sum _{a\;{\text{root of}}\;f}\alpha ^{i}\right).$

Newtons identities allow to express the power sums of the roots of f, and thus above sums, in terms of the coefficients of f. Using Newton's identity back gives thus the coefficients of g in terms of the coefficients of f.

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 d which eliminates the term of degree 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 "principal" or Bring-Jerrard normal form with terms of degree 5,1 and zero.

References

• Adamchik, Victor S.; Jeffrey, David J. (2003). "Polynomial transformations of Tschirnhaus, Bring and Jerrard" (PDF). SIGSAM Bull. 37 (3): 90–94. Zbl 1055.65063.