Difficulties with resultants and Tschirnhaus transformations
I'm experimenting with some polynomial equations, and as a test I'm working through the example given here:
Basically, in order to eliminate the first two terms of the quintic $x^5-x^4-x^3-x^2-x-1$ we need to find a transformation $y=x^2+ax+b$ (such a polynomial transformation is called a Tschirnhaus transformation, after its first discoverer), for which $a$ and $b$ have the effect of producing a quintic equation in $y$ but without $y^4$ or $y^3$ terms. This is as far as I've got so far:
R.<a,b> = QQ S.<y> = R T.<x> = S p = x^5-x^4-x^3-x^2-x-1 res = p.resultant(y-x^2-a*x-b) rc = res.coefficients() solve([rc[-2],rc[-3]],[a,b]) TypeError: a is not a valid variable.
Drat! So I have two questions: why is
a not a valid variable, and why can I not isolate individual coefficients of
res? I can obtain all the coefficients of
res, but I can't isolate just one, with something like