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6 votes

Why do the roots of $\int_0^x (1-s^2)^n ds$ lie on a lemniscate?

Fig. 1 : The roots of polynomial $f_{31}$ and the standard lemniscate with equ. $(x^2+y^2)^2=2(x^2-y^2)$. This figure has been drawn with a "SAGE" program (see remark at the bottom). The ...
Jean Marie's user avatar
3 votes

For a curve to be smooth, it is necessary that its derivative is never equal to $0$. Why? (Complex Analysis, Curves in the complex plane)

I think you are asking for the intuition behind this definition of a smooth curve. Maybe this will help: Imagine you a riding a bike which has a device so that it is painting a line continuously on a ...
Chris L-M's user avatar
2 votes
Accepted

Parameters behind non-symmetric Lissajous loop?

To rephrase your question: find two functions of $\theta$, $f,g$ such that the trace $[x,y]=[f(\theta),g(\theta)]$ traces out your curve. The quick and dirty solution is to read off the functions $f,g$...
user619894's user avatar
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2 votes
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Locus of 4 colinear points in $\mathbb{P}^2$

The intersection $\psi_{1,2,3} \cap \psi_{1,2,4}$ has an extra codim-$2$ component: $\{p_1 = p_2\}$. So you need to subtract the diagonal class supported on those coordinates, ie $\alpha^2 + \alpha \...
ronno's user avatar
  • 12k
2 votes
Accepted

For a curve to be smooth, it is necessary that its derivative is never equal to $0$. Why? (Complex Analysis, Curves in the complex plane)

The condition simply excludes possibility, that the velocity assumes a value of zero and reverses sign possibly twice or some multiples of 2. With a non-monotone parametrization, parts of the curve ...
Roland F's user avatar
  • 2,563
1 vote
Accepted

Inclusion of set $A$ in set $B$ or vice versa, where A and B are defined as follows:

You can give a counterexample: such pair $(x,y)$ that $x^2+y^4\gt1$, $x^4+y^6\le1.$ It’s clear that it’s better to search for such pair along the line $x=y$. So we get $$x^2+x^4\gt1$$ $$x^4+x^6\le1.$$ ...
Aig's user avatar
  • 5,516
1 vote
Accepted

Elliptic curve point "division" by an integer

This is not true in general. For example, let $C$ be any elliptic curve over a number field of rank $r>0$, then the group of rational points is isomorphic to $\Bbb Z^r\times T$, where $T$ is some ...
Lukas Heger's user avatar
  • 21.6k
1 vote

A problem about morphisms from a genus 2 curve to a quartic curve.

The preimage of a line in $\mathbb{P}^2$ is a divisor from the linear system that gives the morphism; its degree is $$ \deg(K + P_1 + P_2) = 4, $$ hence the image is a quartic curve (unless the ...
Sasha's user avatar
  • 17.5k

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