From 6c3fc0d50e7c079ddf88629d5d72ed75b05c4d6a Mon Sep 17 00:00:00 2001
From: jgarnek <jgarnek@amu.edu.pl>
Date: Sat, 23 Sep 2023 10:57:22 +0000
Subject: [PATCH] readme - first content

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+# SAGEMATH module: superelliptic curves and their Artin-Schreier covers
+
+## Basic information
+
+## Usage
+
+The main file is init.sage. In order to use it, type:
+
+```sage: load('init.sage')```
+
+The main two "packages" are intended for:
+- superelliptic curves,
+- $(\mathbb Z/p)^n$-covers of superelliptic curves.
+
+## Superelliptic curves
+
+In order to define a superelliptic curve $C : y^4 = x^6 + 1$ over the finite field with 9 elements,
+use the following commands:
+
+```
+F.<a> = GF(9, 'a')
+Rx.<x> = PolynomialRing(F)
+f = x^6 + 1
+C = superelliptic(f, 4)
+```
+There are three auxilliary classes: superelliptic_function (for functions defined on superelliptic curves), superelliptic_form (for forms defined on superelliptic curves) and superelliptic_cech (for cech cocycles for the de Rham cohomology on superelliptic curves).
+
+For example, in order to define the function $x + y$ on our curve $C$ we can define it like this:
+
+```
+Rxy.<x, y> = PolynomialRing(F, 2)
+fct = superelliptic_function(C, x + y)
+```
+or simpler:
+```
+fct = C.x + C.y
+```
+
+Similarly, in order to define the form $\omega = y \cdot dx$ we may use:
+```
+omega = superelliptic_form(C, y)
+```
+or simpler:
+```
+omega = C.y * C.dx
+```
+
+
+
+## Troubleshooting
+
+- precision
+- threshold
+- no root in the field
+- basis -- coordinates.
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