
06389089
j
2015a.00645
Haggar, Farid
Roots and all: an economical algorithm.
Parabola 50, No. 2, 1117 (2014).
2014
AMT Publishing, Australian Mathematics Trust, University of Canberra, Canberra; School of Mathematics \& Statistics, University of New South Wales, Sydney
EN
H20
H70
monic polynomials
coefficients
sums of powers of roots
identities
recursion relations
simultaneous linear equations
partition function
number of integer solutions
difference equations
partial derivatives
integrals
algebra
From the text: Every polynomial of degree $n$ can be expressed as a product of factors. It is possible to find identities relating the sums of the powers of the roots defined by $s_i=\sum_j (\alpha_j)^i i=1,2,3,\dots$, to the coefficients, $\omega_j$, of the monic polynomial, without actually finding the roots $\alpha_k$ explicitly. It is straightforward to find recursion relations linking $\omega_i$ and $s_i$. We can then use the recursion relations to obtain $s_i$ as functions of $\omega_1$, $\omega_2$,\dots, $\omega_i$ as functions of $s_1$, $s_2$,\dots, $s_i$. The process is laborious. We now describe a more economical way to obtain these expressions for $\omega_i$.