Based on MVLookup.

Page 8. Let’s see an example.

Let’s say the polynomial is $P(X_1, X_2, X_3) = X_1^{2} + 3 X_{2} X_{1} + X_{3}^3$ Let’s say at step $1$, the random element was $1$.

At step 2, the prover will send the coefficients of the polynomial:

\[\begin{align} s_{2}(X) & = \sum_{x \in \{1, -1\}}P(1, X, x) \\ & = p(1, X, 1) + p(1, X, -1) \\ & = 1 + 3X + 1 + 1 + 3X - 1 \\ & = 6 X + 2 \end{align}\]

The coefficients are $6$ and $2$, and the degree is $1$.

The verifier will check that $s_{1}(1) = s_{2}(1) + s_{2}(-1)$, which corresponds to the equality check $s_{i - 1}(r_{i - 1}) = s_{i}(1) + s_{i}(-1)$.

TODO: compute

\[\begin{align} s_{1}(X) & = \sum_{(x_{1}, x_{2}) \in \{\pm 1\}^{2}} P(X, x_{1}, x_{2}) \\ & = P(X, 1, 1) + P(X, 1, -1) + P(X, -1, 1) + P(X, -1, -1) \end{align}\]