I have encountered a problem in my research where I need to prove that all the coefficients of a certain multinomial of the form $R_k(\mathbf{x}, \mathbf{y}) = \frac{P_k(\mathbf{x}, \mathbf{y})}{Q_k(\mathbf{x}, \mathbf{y})}$ are positive, where $\mathbf{x} = (x_1,\dots,x_n)$, and $\mathbf{y} = (y_1,\dots,y_n)$. It is known that $P_k$ is divisible by $Q_k$. Here $k$ can be any positive integer, i.e. I need to prove the statement for each value of $k$. We also know the following about $R_k$:
- $R_k$ is a homogeneous multinomial of degree $n(n-1)(k-1)$,
- $R_k$ is symmetric w.r.t. $\mathbf{x}$, and w.r.t. $\mathbf{y}$ separately,
- $R_k(\mathbf{x}, \mathbf{y}) = R_k(\mathbf{y}, \mathbf{x})$.
I should state that I have no prior background in algebraic geometry, which I suspect might be the kind of tools I need for this problem. So my question to the community is whether anyone has seen similar problems before, or knows of general techniques that are used to solve similar problems, and point me to them. My search so far has been quite futile in trying to find techniques to solve my problem.