Permutation of coefficients of polynomials

Question

Are there any known results relating to permutation of coefficients of polynomials?

for example given a polynomial, if the coefficients are permuted, then are there any results relating the two?

related question, given set of all polynomials that are permutations of coefficients, are there any known results?

The only possible example that I can think of is that if roots of a polynomial are not rational, then all polynomials permutations of it's coefficients also have no rational roots. Not sure if this is correct or incorrect.

Answer 1

Suppose $\;w\neq0\;$ is a root of $\;a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\;$ (All this over some field), then:

$$a_nw^n+a_{n-1}w^{n-1}+\ldots+a_1w+a_0=0\stackrel{\cdot\,w^{-n}}\implies $$

$$a_0(w^{-1})^n+a_1(w^{-1})^{n-1}+\ldots+a_{n-1}w^{-1}+a_n=0\implies w^{-1}\;\;\text{root of}\;\;$$

$$a_0x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n$$

But for the above, I don't know any other more or less significant general relation, though of course there are the polynomials in $\;n\;$ unknowns which remain fixed when a permutation is applied on their coefficients. These are known as symmetric polynomials, but it is not perhaps what you want.

Answer 2

AS an example where your assertion is false: $x^2+x+2$ has non rational roots but $x^2 + 2x + 1 = (x+1)^2$. Permuting the coefficients of a polynomial can change the properties of a polynomial drastically: irreducibility, dicriminant, etc. An exception can be made for polynomials that satisfy the Eisenstein criterium for irreducibility on the condition that the first and last coefficients remain unchanged.