Polynomial decomposition with the aid of the Lagrange-Bürmann formula

Source abstract

The representation of a polynomial (𝑥) as (𝑥) = 𝑔(ℎ(𝑥)), where and ℎ are polynomials, is called the decomposition of the polynomial (𝑥). Not every polynomial has the property of decomposition. The applications of polynomial decomposition are listed. Three new algorithms for decomposing polynomials in one and two variables are presented. One of these algorithms uses the Lagrange-Bürmann formula, which allows for the expansion of one function into powers of another function. The second algorithm uses the method of indeterminate coefficients and the Gröbner basis to decompose a monic polynomial of degree four. The third algorithm uses the Lagrange-Poincaré decomposition formula to decompose a bivariate polynomial. Examples are provided to illustrate each algorithm.