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In mathematics, a superelliptic curve is a plane curve with an equation of the form

$y^{m}=f(x),$

where the exponent $m$ is fixed and $f$ is a polynomial. The case $m=2$ is a hyperelliptic curve: the case $m=3$ is a trigonal curve.

The Diophantine problem of finding integer points on a superelliptic curve can be solved by a method similar to one used for the resolution of hyperelliptic equations: a Siegel identity is used to reduce to a Thue equation.

Definition

More generally, a superelliptic curve is a cyclic branched covering

$C\to \mathbb {P} ^{1}$

of the projective line of degree $m\geq 2$ coprime to the characteristic of the field of definition. The degree $m$ of the covering map is also referred to as the degree of the curve. By cyclic covering we mean that the Galois group of the covering (i.e., the corresponding function field extension) is cyclic.

The fundamental theorem of Kummer theory implies Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. that a superelliptic curve of degree $m$ defined over a field $k$ has an affine model given by an equation

$y^{m}=f(x)$

for some polynomial $f\in k[x]$ of degree $m$ without repeated roots, provided that $C$ has a point defined over $k$, that is, if the set $C(k)$ of $k$-rational points of $C$ is not empty. For example, this is always the case when $k$ is algebraically closed. In particular, function field extension $k(C)/k(x)$ is a Kummer extension.

Ramification

Let $C:y^{m}=f(x)$ be a superelliptic curve defined over an algebraically closed field $k$, and $B'\subset k$ denote the set of roots of $f$ in $k$. Define set

$B={\begin{cases}B'&{\text{ if }}m{\text{ divides }}\deg(f),\\B'\cup \{\infty \}&{\text{ otherwise.}}\end{cases}}$

Then $B\subset \mathbb {P} ^{1}(k)$ is the set of branch points of the covering map $C\to \mathbb {P} ^{1}$ given by $x$.

For an affine branch point $\alpha \in B$, let $r_{\alpha }$ denote the order of $\alpha$ as a root of $f$. Then

$e_{\alpha }={\frac {m}{(m,r_{\alpha })}}$

is the ramification index $e(P_{\alpha ,i})$ at each of the $(m,r_{\alpha })$ ramification points $P_{\alpha ,i}$ of the curve lying over $\alpha \in \mathbb {A} ^{1}(k)\subset \mathbb {P} ^{1}(k)$ (that is actually true for any $\alpha \in k$).

For the point at infinity,

$e_{\infty }={\frac {m}{(m,\deg(f))}}$

is the ramification index $e(P_{\infty ,i})$ at the $(m,\deg(f))$ points $P_{\infty ,i}$ that lie over $\infty$. In particular, the curve is unramified over infinity if and only if its degree $m$ and the degree of the defining polynomial $f$ are relatively prime.

See also

• Hyperelliptic curve
• Artin-Schreier curve
• Kummer theory
• Superellipse

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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