45_plonky3

45_Plonky3.md

@@ -665,9 +665,9 @@ The prover in FRI handles a batch of polynomials and needs to evaluate them over
 
 2. **Batched Quotient Polynomial**: For each distinct degree $d$, the prover computes a batched quotient polynomial as follows:
 
-   $$
-   q)d(\omega^i) = \sum_{j, k} \alpha^{k+j} \cdot \frac{p_j(\omega^i) - y_{jk}}{\omega^i - z_k}
-   $$
+   <div style="text-align: center;">
+   <img src="./images/45_01_quotient.png" alt="Image 1" width="50%" style="display: inline-block;">
+   </div>
 
    This results in a set of batched quotient polynomials for all degrees $d \in S$.
 
@@ -955,15 +955,15 @@ Once the reduced polynomials $\text{reduce}\_0(X)$ and $\text{reduce}\_1(X)$ are
    - These openings yield intermediate values used to compute the reduced polynomials.
 2. **Constructing the Reduced Polynomial**:
    - Compute the reduced trace polynomial:
-     \[
-     \text{reduce}_0(X) = \sum \alpha^i \cdot ldt_0(X)\_i + \alpha^{i+1} \cdot ldt_0'(X)
-     \]
-     where $ldt_0(X)$ and $ldt_0'(X)$ are the Lagrange polynomials formed from the trace evaluations at $\zeta$ and $\zeta_{\text{next}}$.
+   <div style="text-align: center;">
+   <img src="./images/45_02_reduce01.png" alt="Image 1" width="50%" style="display: inline-block;">
+   </div>
+    where $ldt_0(X)$ and $ldt_0'(X)$ are the Lagrange polynomials formed from the trace evaluations at $\zeta$ and $\zeta_{\text{next}}$.
    - Similarly, compute the reduced quotient polynomial:
-     \[
-     \text{reduce}\_1(X) = \sum \alpha^i \cdot ldt_2(X)\_i + \alpha^{i+1} \cdot ldt_3(X)
-     \]
-     where $ldt_2(X)$ and $ldt_3(X)$ are Lagrange polynomials derived from quotient evaluations.
+   <div style="text-align: center;">
+   <img src="./images/45_03_reduce02.png" alt="Image 1" width="50%" style="display: inline-block;">
+   </div>
+    where $ldt_2(X)$ and $ldt_3(X)$ are Lagrange polynomials derived from quotient evaluations.
 3. **Performing the Low-Degree Test**:
    - Both reduced polynomials $\text{reduce}\_0(X)$ and $\text{reduce}\_1(X)$ undergo a low-degree test to verify their degree bounds.