From ddcac21be1e2dd829da09201e417551fa969fb0e Mon Sep 17 00:00:00 2001
From: David Florness <david@florness.com>
Date: Sun, 9 Feb 2020 21:16:47 -0700
Subject: [PATCH] OreSec talk: fix sum expansions

---
 doc/oresec-talk/tallyard.tex | 20 ++++++++++----------
 1 file changed, 10 insertions(+), 10 deletions(-)

diff --git a/doc/oresec-talk/tallyard.tex b/doc/oresec-talk/tallyard.tex
index 99ca40a..ed98e17 100644
--- a/doc/oresec-talk/tallyard.tex
+++ b/doc/oresec-talk/tallyard.tex
@@ -235,10 +235,10 @@
   \begin{itemize}
   \item Everyone now has $S_1$, $S_2$, $\dots$, $S_k$: \pause
     \begin{align*}
-      S_1 &= \sum_{j=1}^k c_j + x_1 \sum_{j=1}^k a_{(j,1)} + x_1^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_1^{k-1} \sum_{j=1}^k a_{(j,2)} \\
-      S_1 &= \sum_{j=1}^k c_j + x_2 \sum_{j=1}^k a_{(j,1)} + x_2^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_2^{k-1} \sum_{j=1}^k a_{(j,2)} \\
+      S_1 &= \sum_{j=1}^k c_j + x_1 \sum_{j=1}^k a_{(j,1)} + x_1^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_1^{k-1} \sum_{j=1}^k a_{(j,k-1)} \\
+      S_1 &= \sum_{j=1}^k c_j + x_2 \sum_{j=1}^k a_{(j,1)} + x_2^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_2^{k-1} \sum_{j=1}^k a_{(j,k-1)} \\
           &\vdots \\
-      S_k &= \sum_{j=1}^k c_j + x_k \sum_{j=1}^k a_{(j,1)} + x_k^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_k^{k-1} \sum_{j=1}^k a_{(j,2)}
+      S_k &= \sum_{j=1}^k c_j + x_k \sum_{j=1}^k a_{(j,1)} + x_k^2 \sum_{j=1}^k a_{(j,2)} + \cdots + x_k^{k-1} \sum_{j=1}^k a_{(j,k-1)}
     \end{align*} \pause
   \item These are $k$ points all on the same $k-1$ degree polynomial:
     \begin{equation*}
@@ -253,17 +253,17 @@
     coefficients and constant term of it with a little linear algebra: \pause
     \begin{equation*}
       \begin{bmatrix}
-        1 & x_1 & x_1^2 & \cdots & x_1^k \\
-        1 & x_2 & x_2^2 & \cdots & x_2^k \\
+        1 & x_1 & x_1^2 & \cdots & x_1^{k-1} \\
+        1 & x_2 & x_2^2 & \cdots & x_2^{k-1} \\
         \vdots & & & \ddots \\
-        1 & x_k & x_k^2 & \cdots & x_k^k
+        1 & x_k & x_k^2 & \cdots & x_k^{k-1}
       \end{bmatrix}
             \begin{bmatrix}
               \sum_{j=1}^k c_j \\
               \sum_{j=1}^k a_{(j,1)} \\
               \sum_{j=1}^k a_{(j,2)} \\
               \vdots \\
-              \sum_{j=1}^k a_{(j,2)}
+              \sum_{j=1}^k a_{(j,k-1)}
             \end{bmatrix} =
             \begin{bmatrix}
               S_1 \\
@@ -279,10 +279,10 @@
   \begin{equation*}
     \text{RREF}\left(
       \begin{bmatrix}
-        1 & x_1 & x_1^2 & \cdots & x_1^k & S_1 \\
-        1 & x_2 & x_2^2 & \cdots & x_2^k & S_2 \\
+        1 & x_1 & x_1^2 & \cdots & x_1^{k-1} & S_1 \\
+        1 & x_2 & x_2^2 & \cdots & x_2^{k-1} & S_2 \\
         \vdots & & & \ddots \\
-        1 & x_k & x_k^2 & \cdots & x_k^k & S_k
+        1 & x_k & x_k^2 & \cdots & x_k^{k-1} & S_k
       \end{bmatrix}
     \right)
   \end{equation*}
-- 
2.38.4