diff --git a/complementarioconica.tex b/complementarioconica.tex index 7f039c8..45cd251 100644 --- a/complementarioconica.tex +++ b/complementarioconica.tex @@ -216,7 +216,7 @@ $(1,e_2)$ & $(1,v_1)$ & $(d_2^{-1},v_2)$ & $(d_2,e_2)$ & $(d_ $(1,e_3)$ & $(1,v_2)$ & $(1,v_1)$ & $(d_2,e_3)$ & $(d_2,v_2)$ & $(d_2,v_1)$ \\\hline $(1,e_4)$ & $(1,v_2)$ & $(d_2^{-1},v_1)$ & $(d_2,e_4)$ & $(d_2,v_2)$ & $(1,v_1)$ \\\hline $(1,e_5)$ & $(1,v_2)$ & $(d_2^2,v_0)$ & $(d_2,e_5)$ & $(d_2,v_2)$ & $(d_2^{-1},v_0)$ \\\hline -$(1,e_6)$ & $(1,v_2)$ & $(1,v_0)$ & $(d_2,e_6)$ & $(d_2,v_2)$ & $(d_2,v_0)$ \\v +$(1,e_6)$ & $(1,v_2)$ & $(1,v_0)$ & $(d_2,e_6)$ & $(d_2,v_2)$ & $(d_2,v_0)$ \\\hline $(1,e_7)$ & $(1,v_1)$ & $(d_2,v_0)$ & $(d_2,e_7)$ & $(d_2,v_1)$ & $(d_2^2,v_0)$ \\\hline $(1,e_8)$ & $(1,v_1)$ & $(d_2^{-1},v_0)$ & $(d_2,e_8)$ & $(d_2,v_1)$ & $(1,v_0)$ \\\hline $(1,e_9)$ & $(1,v_1)$ & $(1,v_0)$ & $(d_2,e_9)$ & $(d_2,v_1)$ & $(d_2,v_0)$ \\\hline diff --git a/generaldegree.tex b/generaldegree.tex index 707725c..c2a1863 100644 --- a/generaldegree.tex +++ b/generaldegree.tex @@ -13,6 +13,7 @@ \usepackage{hyperref} \newtheorem{theorem}{Theorem} +\newtheorem{example}{Example} \newtheorem{lemma}{Lemma} \newtheorem{definition}{Definition} \newcommand{\CC}{\mathbb{C}} @@ -291,4 +292,30 @@ In particular, the $d^3$ vertices of the simplicial complex $\Delta^d$ will be i The edges of $\Delta$ can be identified with the lines of $S_1$ parametrized as $(1-t,t,0)$, $(1-t,0,t)$ and $(0,t,1-t)$ with $t\in[0,1]$. Then, the preimages of these lines by $\phi$ are of the form $(\sqrt[d]{1-t}\xi_i, \sqrt[d]{t}\xi_j, 0)$ in the first case and analogous in the rest, where we denote $\sqrt[s]{x}$ the positive real $d$-root for $x$ real, and where $\xi_i$ and $\xi_j$ are two any $d$-$th$ roots of unity. For each of these preimages we can define an edge between the corresponding vertices in $\Delta^d$. The $2$-faces of the simplicial complex $\Delta^d$ will then be identified to the areas between the lines $(\sqrt[d]{1-t}\xi_i, \sqrt[d]{t}\xi_j, 0)$, $(\sqrt[d]{1-t}\xi_i,0, \sqrt[d]{t}\xi_k)$ and $(0,\sqrt[d]{t}\xi_j, \sqrt[d]{1-t}\xi_k)$. + +\begin{example}{} + We are going to compute now the simplicial complex for $d=2$, this is, $S_2=\{(x,y,z)\in M_2| x^2, y^2, z^2\in\RR_{\geq 0}\}$. + + First, we can observe that, from all points in $\CC$, the only ones whose square power is real are either the real numbers of the pure imaginary numbers. And from these, the later have negative square power. Therefore, we can conclude that $S_2=\{(x,y,z)\in\RR^3| x^2+y^2+z^2=1\}$, that is, $S_2$ is the real three dimensional sphere. We now see the simplicial complex $\Delta^2$ that can be associated to it such that $\phi(\Delta^2)=\Delta$. + + The vertices of $\Delta$, $v_0$, $v_1$ and $v_2$, can be identified to $(1,0,0)$, $(0,0,1)$ and $(0,1,0)$ respectively. The preimages of these points are: + \begin{equation} + \begin{array}{rll} + \phi^{-1}((1,0,0))&=&\{(1,0,0),(-1,0,0)\}\\ + \phi^{-1}((0,1,0))&=&\{(0,1,0),(0,-1,0)\}\\ + \phi^{-1}((0,0,1))&=&\{(0,0,1),(0,0,-1)\} + \end{array} + \end{equation} + + The edges of the simplex, $[v_0v_1]$, $[v_0v_2]$ and $[v_1v_2]$, can be identified to the points of $S_1$ given respectively by $\{(1-t, 0, t)|t\in[0,1]\}$, $\{(1-t,t,0)|t\in[0,1]\}$ and $\{(0,t,1-t)|t\in[0,1]\}$. Therefore, each edge will have four preimages. In the first case, these preimages are: + \begin{equation} + \phi^{-1}((1-t,t,0))=\left\{\begin{array}{l} + \{(\sqrt{1-t},\sqrt{t},0)|t\in[0,1]\}\\ + \{(\sqrt{1-t},-\sqrt{t},0)|t\in[0,1]\}\\ + \{(-\sqrt{1-t},\sqrt{t},0)|t\in[0,1]\}\\ + \{(-\sqrt{1-t},-\sqrt{t},0)|t\in[0,1]\} + \end{array}\right. + \end{equation} + For the rest of the edges the preimages by $\phi$ would be analougous. +\end{example} \end{document} \ No newline at end of file