If $P$ is a real quadratic polynomial, then $|P(z)| geq sqrt{Delta} |Im(z)|$
$begingroup$
I was reading an article which contains a statement about the distance in the hyperbolic plane: if $gamma in SL_2(mathbb R)$ is hyperbolic then $inf_{z in H} d(gamma z, z) = arcosh(Tr(gamma)^2/2-1)$. This implies the following:
If $P$ is a real quadratic polynomial with real roots and discriminant $Delta$, then $inf_{z in mathbb C - mathbb R} |P(z)|/|Im(z)| = sqrt{Delta}$.
I suppose I could prove this by doing a long computation if I wanted to. But I'd like to know if there is a conceputal proof that explains where the discriminant comes from.
polynomials complex-numbers hyperbolic-geometry discriminant
asked Jan 11 at 10:37
bartobarto
13.7k32682
$endgroup$
add a comment |
$begingroup$
I was reading an article which contains a statement about the distance in the hyperbolic plane: if $gamma in SL_2(mathbb R)$ is hyperbolic then $inf_{z in H} d(gamma z, z) = arcosh(Tr(gamma)^2/2-1)$. This implies the following:
If $P$ is a real quadratic polynomial with real roots and discriminant $Delta$, then $inf_{z in mathbb C - mathbb R} |P(z)|/|Im(z)| = sqrt{Delta}$.
I suppose I could prove this by doing a long computation if I wanted to. But I'd like to know if there is a conceputal proof that explains where the discriminant comes from.
polynomials complex-numbers hyperbolic-geometry discriminant
asked Jan 11 at 10:37
bartobarto
13.7k32682
$endgroup$
add a comment |
$begingroup$
I was reading an article which contains a statement about the distance in the hyperbolic plane: if $gamma in SL_2(mathbb R)$ is hyperbolic then $inf_{z in H} d(gamma z, z) = arcosh(Tr(gamma)^2/2-1)$. This implies the following:
If $P$ is a real quadratic polynomial with real roots and discriminant $Delta$, then $inf_{z in mathbb C - mathbb R} |P(z)|/|Im(z)| = sqrt{Delta}$.
I suppose I could prove this by doing a long computation if I wanted to. But I'd like to know if there is a conceputal proof that explains where the discriminant comes from.
polynomials complex-numbers hyperbolic-geometry discriminant
asked Jan 11 at 10:37
bartobarto
13.7k32682
$endgroup$
I was reading an article which contains a statement about the distance in the hyperbolic plane: if $gamma in SL_2(mathbb R)$ is hyperbolic then $inf_{z in H} d(gamma z, z) = arcosh(Tr(gamma)^2/2-1)$. This implies the following:
If $P$ is a real quadratic polynomial with real roots and discriminant $Delta$, then $inf_{z in mathbb C - mathbb R} |P(z)|/|Im(z)| = sqrt{Delta}$.
I suppose I could prove this by doing a long computation if I wanted to. But I'd like to know if there is a conceputal proof that explains where the discriminant comes from.
polynomials complex-numbers hyperbolic-geometry discriminant
polynomials complex-numbers hyperbolic-geometry discriminant
asked Jan 11 at 10:37
bartobarto
13.7k32682
asked Jan 11 at 10:37
bartobarto
13.7k32682
edited Jan 11 at 11:48
barto
asked Jan 11 at 10:37
bartobarto
13.7k32682
asked Jan 11 at 10:37
bartobarto
13.7k32682
asked Jan 11 at 10:37
bartobarto
13.7k32682
13.7k32682
add a comment |
add a comment |
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