I have a system of linear equations given by $$\mathbf{y}=\mathbf{Ax}$$ where $\mathbf{A}$ is a $m \times n$ matrix and $\mathbf{x}$ and $\mathbf{y}$ are vectors with $n$ and $m$ entries, respectively.
I am trying to find a simple way to compute $$\prod_{i=1}^{m} y_i^{k_i},$$ where $y_i$ is the $i-$th element of the $\mathbf{y}$ vector and $(k_1,k_2,\ldots,k_m)$ are integers, as function of $\mathbf{A}$ and $\mathbf{x}$ elements.
To compute $y_i^{k_i}$ we can use the multinomial theorem, $$y_i^{k_i}=\left(\sum_{j=1}^n a_{ij}x_j\right)^{k_i}=\sum _{s_{1}+s_{2}+\cdots +s_{n}=k_i}{k_i \choose s_{1},s_{2},\ldots ,s_{n}}\prod _{t=1}^{n}(a_{it}x_{t})^{s_{t}},$$ where $${k_i \choose s_{1},s_{2},\ldots ,s_{n}}=\frac{k_i!}{s_{1}!s_{2}!\ldots s_{n}!}$$ is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices $s_1$ through $s_n$ such that the sum is $k_i$. However, the expression is already very complicated and the product $\prod_{i=1}^{m} y_i^{k_i}$ would be painfully long to express as a function of $\mathbf{A}$ and $\mathbf{x}$ elements.
Is there a simpler way to express $\prod_{i=1}^{m} y_i^{k_i}$ ?