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Let $m_\lambda(X_1,X_2,...X_N)$ be a monomial symmetric function with partition $\lambda$.

For example:

$ m_{(3,1,1)}(X_1,X_2,X_3) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $

Is there a general formula for roots of $m_\lambda$ if $X_j$ is restricted to elements of $\mathbb C$ with $\| X_j \|=1$?

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    The usual term is not a "symmetrical monomial", it is a "monomial symmetric function".2011-11-13

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I doubt it. Let's look at $m_{(1,0,0,0,0)}$. You want the solutions of $a+b+c+d+e=0$ with all variables on the unit circle. It will be hard enough to find a formula for that special case, much less for the general case.

Note that $m_{(1,0)}$ is $a+b=0$ which is solved by $a=e^{it}$, $b=e^{i(t+\pi)}$. Then $m_{(1,0,0)}$ is $a+b+c=0$, which forces $a,b,c$ to be vertices of an equilateral triangle. Next, $m_{(1,0,0,0)}$ is $a+b+c+d=0$, and with a bit of work you can show that $a,b,c,d$ must be vertices of a rectangle. But once you get up to 5 unknowns the geometric argument doesn't give you anything that simple.

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    hm, right, maybe I was confused by the squares...2011-11-22