How do I go from LPC coefficients to a filter polynomial?
The prediction polynomial is the representation in the $z$-domain of the first equation you wrote.

When performing LPC, one assumes that the filter consists of only poles and the transfer function is $\frac{1}{A(z)}$ where

$$A(z)=1 - \sum_{i=1}^{M} a_i z^{-i}$$

That's the polynomial you are talking about. Note that it should have only negative powers of $z$, not positive.

As you wrote in your question, one estimates the signal $h(n)$ based on past values of that same signal. The $z^{-i}$ factors in the polynomial represent those delays in the $z$-domain, while the coefficients $a_i$ are just constants and, due to the $z$-transform linearity, they just stay as they are.