started analyzing t

documents/math-minimal-distance-to-cubic-function/analyzing-t.tex

@@ -0,0 +1,14 @@
+\begin{align}
+    t &:= \sqrt[3]{\sqrt{3 \cdot (4 \alpha^3 + 27 \beta^2)} -9\beta}\\
+    &= \sqrt[3]{\sqrt{3 \cdot \left (4 \left (\frac{1- 2 aw}{2 a^2} \right )^3 + 27 \left (\frac{-z}{2 a^2} \right )^2 \right )} -9 \frac{-z}{2 a^2}}\\
+    &= \sqrt[3]{\sqrt{3 \cdot \left (4 \left (\frac{1- 2 a (y_P+\frac{b^2}{4a}-c)}{2 a^2} \right )^3 + 27 \left (\frac{-(x_P+\frac{b}{2a})}{2 a^2} \right )^2 \right )} 
+    -9 \frac{-(x_P+\frac{b}{2a})}{2 a^2}}\\
+    &= \sqrt[3]{\sqrt{12a^4 \cdot \left (4 \frac{\left ( 1- 2 a (y_P+\frac{b^2}{4a}-c) \right )^3}{2 a^2}  + 27 \left (x_P^2+2 x_P \frac{b}{2a} + \frac{b^2}{4a^2} \right )\right )}
+    + 9 \frac{x_P+\frac{b}{2a}}{2 a^2}}\\
+    &= \sqrt[3]{\sqrt{\frac{12a^4}{4a^2} \left (8 \left ( 1- 2 a (y_P+\frac{b^2}{4a}-c) \right )^3  + 27 (4 a^2 x_P^2+4a x_P \frac{b}{2a} + b^2 )\right )}
+    + 9 \frac{x_P+\frac{b}{2a}}{2 a^2}}\\
+    &= \sqrt[3]{\sqrt{3a^2 \left (8 \left ( 1- 2 a (y_P+\frac{b^2}{4a}-c) \right )^3  + 27 (4 a^2 x_P^2+4a x_P \frac{b}{2a} + b^2 )\right )}
+    + 9 \frac{x_P+\frac{b}{2a}}{2 a^2}}
+\end{align}
+
+\todo[inline]{When is $t = 0$? When is $t \in \mdr$?}

documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex

@@ -164,6 +164,10 @@ Otherwise, there is only one solution $x_1 = 0$.
 Let $t$ be defined as
 \[t := \sqrt[3]{\sqrt{3 \cdot (4 \alpha^3 + 27 \beta^2)} -9\beta}\]
 
+\subsubsection{Analyzing $t$}
+\input{analyzing-t.tex}
+
+\subsubsection{Solutions of $x^3 + \alpha x + \beta$}
 I will make use of the following identities:
 \begin{align*}
     (1-i \sqrt{3})^2     &= -2 (1+i \sqrt{3})\\