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The $k$th elementary symmetric polynomial adds every product obtained by choosing $k$ distinct positions from $x_1,\dots ,x_n$:The conventions are $e_0=1$ and $e_k=0$ for $k>n$. The positions must be distinct, but their values may coincide. Each permutation of the variables merely rearranges the summands, so this is a symmetric polynomial.

For $x_1=1$, $x_2=2$, $x_3=3$, we obtain $e_1=1+2+3=6$, $e_2=1\cdot 2+1\cdot 3+2\cdot 3=11$, and $e_3=1\cdot 2\cdot 3=6$. Expanding the product givesIn general, ${\prod }_{i=1}^n(z-x_i)=z^n-e_1{z}^{n-1}+e_2{z}^{n-2}-\cdots +(-1{)}^n{e}_n$. Thus the coefficients of a monic polynomial collect elementary symmetric polynomials of its roots of a polynomial, with alternating signs. These are the Vieta formulas.^[ref]

The boundary can recover more than a count. Under the hypotheses of the argument principle, let $h$ be a holomorphic function on a neighborhood of the contour and its interior. If the enclosed zeros of a function are $a_j$ with zero multiplicities $m_j$ and the enclosed poles are $b_k$ of orders $n_k$, thenMultiplying the local logarithmic derivative by $h$ changes its residue from $m_j$ to $m_{j}h(a_j)$, or from $-n_k$ to $-n_{k}h(b_k)$. The residue theorem then gives the identity. Choosing $h=1$ recovers the argument principle; choosing $h(z)=z$ gives the sum of the enclosed zero locations minus the sum of the enclosed pole locations, both counted with zero multiplicities or pole orders.^[ref]

For the rational function in argument principle for a rational function, the $h(z)=z$ integral is $2(1/4)-(-1/2)=1$. Adding the known pole location $-1/2$ back gives a zero-location sum of $1/2$. Dividing by the zero count two gives the arithmetic mean of the enclosed zeros, with multiplicity, as $1/4$. Choosing $h(z)=z^k$ similarly gives the difference between the power-sum symmetric polynomials evaluated on the enclosed zero and pole locations, repeating each location according to its multiplicity or order. These identities let boundary information constrain both the number and aggregate locations of enclosed zeros of a function.