ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 01 Jan 2021 17:40:17 +0100Computing power series local coordinates on an algebraic curvehttps://ask.sagemath.org/question/54184/computing-power-series-local-coordinates-on-an-algebraic-curve/Let's say I have an elliptic curve $E$ given by a Weierstrass equation in $x$ and $y$. Let's say I choose a uniformizer $t$ around a point $P$ on $E$.
Then is there a SAGE function that writes $x$ or $y$ as a power series in $t$ in a neighborhood of $P$?
Often, I might take something like $t=x$, so it's a matter of writing $y$.
Even more to the point, I want to take a differential form regular at $P$, say in the form $dx/y$ or $xdx/y$, and write it as a power series times $dt$?
More generally, for any smooth point on a curve, or even any smooth point on a scheme and a system of parameters at that point, there should be a function that takes a regular function in a neighborhood of that point and writes it as a power series in those parameters. I could imagine writing something like this myself, but it would take a lot of work, and I would hope this is already implemented.Tue, 10 Nov 2020 04:51:17 +0100https://ask.sagemath.org/question/54184/computing-power-series-local-coordinates-on-an-algebraic-curve/Answer by slelievre for <p>Let's say I have an elliptic curve $E$ given by a Weierstrass equation in $x$ and $y$. Let's say I choose a uniformizer $t$ around a point $P$ on $E$.</p>
<p>Then is there a SAGE function that writes $x$ or $y$ as a power series in $t$ in a neighborhood of $P$?</p>
<p>Often, I might take something like $t=x$, so it's a matter of writing $y$.</p>
<p>Even more to the point, I want to take a differential form regular at $P$, say in the form $dx/y$ or $xdx/y$, and write it as a power series times $dt$?</p>
<p>More generally, for any smooth point on a curve, or even any smooth point on a scheme and a system of parameters at that point, there should be a function that takes a regular function in a neighborhood of that point and writes it as a power series in those parameters. I could imagine writing something like this myself, but it would take a lot of work, and I would hope this is already implemented.</p>
https://ask.sagemath.org/question/54184/computing-power-series-local-coordinates-on-an-algebraic-curve/?answer=55054#post-id-55054You might be interested in this package that can be installed on top of Sage:
- [the abelfunctions package for SageMath](https://github.com/abelfunctions/abelfunctions)
Fri, 01 Jan 2021 17:40:17 +0100https://ask.sagemath.org/question/54184/computing-power-series-local-coordinates-on-an-algebraic-curve/?answer=55054#post-id-55054