id: 06389089
dt: j
an: 2015a.00645
au: Haggar, Farid
ti: Roots and all: an economical algorithm.
so: Parabola 50, No. 2, 11-17 (2014).
py: 2014
pu: AMT Publishing, Australian Mathematics Trust, University of Canberra,
Canberra; School of Mathematics \& Statistics, University of New South
Wales, Sydney
la: EN
cc: H20 H70
ut: 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
ci:
li:
ab: 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 $s_i=\sum_j (α_j)^i i=1,2,3,\dots$, to
the coefficients, $ω_j$, of the monic polynomial, without actually
finding the roots $α_k$ explicitly. It is straightforward to find
recursion relations linking $ω_i$ and $s_i$. We can then use the
recursion relations to obtain $s_i$ as functions of $ω_1$,
$ω_2$,\dots, $ω_i$ 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 $ω_i$.
rv: