reduce proof size by using stacked PCS and jagged PCS

[kunxian-xia]

Let's say we want to open many multilinear polynomials at a single point, we can stack them into a new "giga" multilinear polynomial by concatenation. Then we can treat/reshape this "giga" multilinear polynomial as several smaller multilinear polynomials with same number of variables.

[kunxian-xia]

Stacked PCS for dense MLE

Let's say we have a giga mle $M(x)$ and we decompose / reshape it into multiple smaller component MLEs $m_j(x)$ with same number of variables h by the formula

$$m_j(b) = M(j, b) = M(j * 2^h + b)$$

Then to open the original giga mle at a point $r = (r_j, r_b)$, this is equivalent to open the component MLEs at point $r_b$ and then check if the evaluations $\{ m_j(r_b) \}$ satisfy the equation

$$M(r) = \sum_j eq(r_j, j) * m_j(r_b)$$

The reason for the above check should work is

$$\begin{align} M(r) & = & \sum_{j, b} M(j, b) * eq(r_j, j) * eq(r_b, b) \\\ & = & \sum_j eq(r_j, j) * \sum_b M(j, b) * eq(r_b, b) \\\ & = & \sum_j eq(r_j, j) * m_j(r_b) \end{align}$$