Is there a criterion for showing that a symmetric ($f(x,y)=f(y,x)$) two variable continuous function $f:\Bbb S^1\times\Bbb S^1\to \Bbb R^2$ pass from the origin? i.e. $$\exists\,(x,y)\in \Bbb S^1\times\Bbb S^1,\,x\neq y\quad s.t. \quad f(x,y)=(0,0).$$ where we know that $f$ pass from all four regions and $f(x,x)=(0,0)$ for all $x\in\Bbb S^1$. I tried to define a real value function from $f$ and use from intermediate value theorem but I couldn't find a helpful map. Is it sufficient to show ${\rm Im} f$ is simply connected?


Let $\alpha(\theta)=\left\lgroup \begin{array}{c} \cos \theta \\ \sin \theta \end{array} \right\rgroup \in \mathbb{S}^1$ and $\beta(\varphi)=\left\lgroup \begin{array}{c} \cos \varphi \\ \sin \varphi \end{array} \right\rgroup\in\mathbb{S}^1$ parametrizations of $\mathbb{S}^1$ whit $0\leq \theta\leq 2\cdot \pi$ and $0\leq \varphi\leq 2\cdot \pi$.

By Poincaré-Miranda theorem if there are $\theta^\prime,\theta^{\prime\prime}\in [0,2\pi]$ and $\varphi^\prime,\varphi^{\prime\prime}\in [0,2\pi]$ such that $$ f_1 \left( \left\lgroup \begin{array}{c} \cos \theta^\prime \\ \sin \theta^{\prime} \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi \\ \sin \varphi \end{array} \right\rgroup \right) <0, \quad f_1 \left( \left\lgroup \begin{array}{c} \cos \theta^{\prime\prime} \\ \sin \theta^{\prime\prime} \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi \\ \sin \varphi \end{array} \right\rgroup \right) >0,\quad \forall \varphi \in [\varphi^{\prime},\varphi^{\prime\prime}] $$

$$ f_2 \left( \left\lgroup \begin{array}{c} \cos \theta \\ \sin \theta \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi^{\prime} \\ \sin \varphi^{\prime} \end{array} \right\rgroup \right) <0, \quad f_2 \left( \left\lgroup \begin{array}{c} \cos \theta \\ \sin \theta \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi^{\prime\prime} \\ \sin \varphi^{\prime\prime} \end{array} \right\rgroup \right) >0,\quad \forall \theta \in [\theta^{\prime},\theta^{\prime\prime}] $$ then there are $\theta^\ast$ and $\varphi^\ast$ such that $$ f_1 \left( \left\lgroup \begin{array}{c} \cos \theta^{\ast} \\ \sin \theta^{\ast} \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi^{\ast} \\ \sin \varphi^{\ast} \end{array} \right\rgroup \right) =0\;\; \mbox{ and } \;\; f_2 \left( \left\lgroup \begin{array}{c} \cos \theta^{\ast} \\ \sin \theta^{\ast} \end{array} \right\rgroup , \left\lgroup \begin{array}{c} \cos \varphi^{\ast} \\ \sin \varphi^{\ast} \end{array} \right\rgroup \right) =0, $$ The Poincaré–Miranda theorem is a generalization of the intermediate value theorem.

  • $\begingroup$ How this theorem guarantee that $\theta^*\neq \varphi^*$? $\endgroup$ – C.F.G Jan 4 '18 at 17:01

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