Question about the permutations of roots as polynomial coefficients

[galoisjr]

Ok, so obviously, given some polynomial P(x) of degree r, it has r roots in the complex numbers by the FTOA, and if these roots are u_1, u_2,... it can be written as

$$\begin{array}{l} P(x) = (x - {u_1})(x - {u_2})(x - {u_3}) \cdots \\ P(x) = {x^r} - ({u_1} + {u_2} + {u_3} + \ldots ){x^{r - 1}} + ({u_1}{u_2} + {u_1}{u_3} + {u_2}{u_3}+ \ldots){x^{r - 2}} - ({u_1}{u_2}{u_3} + \ldots ){x^{r - 3}} + \ldots \end{array}$$

Obviously the coefficient of the r-n power is the sum over the permutations of the r roots taken n at a time.

My question is:

Is there a shorthand notation for referring to this in an equation, i.e. a combinatorial expression in terms of the discrete set of roots?

Obviously when talking about just plain numbers we can refer to the standard nCr and nPr formulas, but in this case we must view the set of roots as a discrete set of objects rather than numbers, hence my dilemma.

 

[micromass]

I don't know of any standard notation. In any case, you will probably like to know about the elementary symmetric polynomials: http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial

So perhaps you could use the notation $e_k(u_1,...,u_n)$...

 

[galoisjr]

That is exactly what I was looking for.

Thank you