Let $f, g$ be a pair of rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil symbol of $f$ and $g$ at $\lambda$ is defined by the following formula: $$ (f,g)_{\lambda}=(-1)^{ab}\frac{f^a}{g^b}(\lambda). $$

Question: For which pairs of rational functions $f, g$ Weil symbol $ (f,g)_{\lambda}$ equals to $\pm1$ at every point $\lambda\in \mathbb{C}?$

It is easy to find some pairs of such functions. For every $r(t)\in \mathbb{C}(t)$ we can take $f=r(t)^a(1-r(t))^b$ and $g=r(t)^c(1-r(t))^d.$

This problem can be generalized to arbitrary Riemann surfaces, but it is probably very hard, because nontrivial examples of such pairs of functions come from $A-$polynomials of some knots.

Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil symbol of $f$ and $g$ at $\lambda$ is defined by the following formula: $$ (f,g)_{\lambda}=(-1)^{ab}\frac{f^a}{g^b}(\lambda). $$

Question: For which pairs of rational functions $f, g$ Weil symbol $ (f,g)_{\lambda}$ equals to $\pm1$ at every point $\lambda\in \mathbb{C}?$

It is easy to find some pairs of such functions. For every $r(t)\in \mathbb{C}(t)$ we can take $f=r(t)^a(1-r(t))^b$ and $g=r(t)^c(1-r(t))^d.$

This problem can be generalized to arbitrary Riemann surfaces, but it is probably very hard, because nontrivial examples of such pairs of functions come from $A-$polynomials of some knots.