Can Any 4th Degree Polynomial Be Expressed as a Quadratic of a Quadratic?

[okkvlt]

Question about quadratics of quadratics

can any 4th degree polynomial be expressed as a quadratic of a quadratic function?


or in the more general case, can any polynomial of degree 2^n be expressed as n-many quadratic functions of quadratic functions?
and given a polynomial of degree 2^n is there a way to find the coefficients of the quadratics?


i tried finding a way to reduce the 4th degree poly into a quadratic of a quadratic. but the main problem is when trying to find the coefficients that i have 5 equations in 6 unknowns, so the system to find the coefficients isn't determined. why and what does this mean?


r=(R4)X^4+(R3)X^3+(R2)X^2+(R1)X+R0

p[q[X]]=P2(Q2*X^2+Q1*X+Q0)^2+P1(Q2*X^2+Q1*X+Q0)+P0

in order for r[x] to equal p[q[x]]:

R4=(P2)(Q2)
R3=2(P2)(Q2)(Q1)
R2=(P2)(2(Q2)(Q0)+(Q1)^2)+(P1)(Q2)
R1=2(P2)(Q0)(Q1)+(P1)(Q1)
R0=(P2)(Q0)^2+(P1)(Q0)+(P0)

im wondering if i could just set one of the coefficients of either quadratic equal to zero. but even then the system is unsolveable by matrices, and I am not sure what terms are allowed to be zero.

 

[symbolipoint]

? Yes...? Are you really asking about composition of functions? If so, then, yes.

 

[gel]

No you can't.

If it was possible then you could always take Q0=0 and Q2=1, just by absorbing these constants into p. Then, you have 5 equations in 4 unknowns.