How to build comparison operator (comparator) in an arithmetic circuit
I believe this is generally done in MPC by converting the sharing of the field element to some sharing more amenable to comparison. See this (section 6.3), or chapter 9 in Secure Multiparty Computation and Secret Sharing by Cramer et al.

Generically, comparisons are dependent on how you represent values in $\mathbb{F}_p$. For example, in $\mathbb{F}_3$ you might want to say that $1 \leq 2$, but since $-1$ is in the same equivalence class as $1$ any polynomial that computes this (for $\mathbb{F}_3\cong \{0,1,2\}$) would also compute $1 \leq -1$ (for $\mathbb{F}_3\cong\{-1,0,1\}$). You could of course use Lagrange Interpolation to get any polynomial to have arbitrary values, but it'd be difficult to call that polynomial "the comparison polynomial", and your model of arithmetic circuits would have to be with respect to a specific representation of field elements (which sounds non-standard, but I'm no expert).

The book by Cramer et al has some commentary on how comparisons using solely $\mathbb{F}_p$ arithmetic leads to large circuits, so maybe there is actually some "comparison polynomial" (and m preceding paragraph is wrong), but it appears the standard thing to do still is to convert to an easier representation to do comparisons on for efficiency.