Given relations of coefficients and $m$ zeros of a complex polynomial, find the polynomial of degree $2n$ and $m \geq n$.

Question

Given relations of coefficients and $m$ zeros of a complex polynomial (coefficients are complex), find the polynomial of degree $2n$ and $m \geq n$. For examples, we are finding $P(x)=C_{2n}x^{2n} + C_{2n-1}x^{2n-1}+\cdots+C_1x+C_0$, where $C_k = C_{2n-k}$, or $C_k = C^*_{2n-k}$ (* means conjugate). Totally, there are $2n$ coefficients to be solved and there are $m$ roots and $n$ relations between coefficients. $m+n\geq 2n$