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$9$. Consider the parametric curve $K\subset R^3$ given by

$$x = (2 + \cos(2s)) \cos(3s)$$

$$y = (2 + \cos(2s)) \sin(3s)$$

$$z = \sin(2s)$$

a) Express the equations ofKas polynomial equations in $x,\ y,\ z,\ a = \cos(s),\ b = \sin(s)$.

Hint: Trig identities.b) By computing a Groebner basis for the ideal generated by the equations from part $a$ and $a^2 + b^2 – 1$ as in Exercise 8, show that

Kis (a subset of) an afﬁne algebraic curve. Find implicit equations for a curve containingK.c) Show that the equation of the surface from Exercise 8 is contained in the ideal generated

by the equations from part b. What does this result mean geometrically? (You can actually reach the same conclusion by comparing the parametrizations ofTandK, without

calculations.)

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- Covering projective variety with open sets $U_i$ such that $\pi^{-1}(U_i) \cong U_i \times \Bbb{A}^1$: How to improve geometric intuition?

I try to solve this problem, on page 102 of Cox’s “Ideal, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra”.

On the first question, I get

$$x=(2+2\cos^2s-1)(4\cos^3s-3\cos s)=(1+2a^2)(4a^3-3a),$$

$$y=(1+2a^2)(3\sin s-4\sin^3 s)=(1+2a^2)(3b-4b^3),$$

$$z=2ab.$$

I was wondering whether they are *right*, since the Groebner basis given by them is extremely bad,

Any comments? Thanks.

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For what it’s worth: you could try easing things a bit for your Gröbner basis computations by using the Weierstrass substitutions

$$\begin{align*}\cos\,s&=\frac{1-u^2}{1+u^2}\\\sin\,s&=\frac{2u}{1+u^2}\end{align*}$$

after of course using multiple angle identities to expand out the trigonometric functions. Since it seems you’re using *Mathematica*, here’s how I’d do it if I were in your shoes:

```
GroebnerBasis[TrigExpand[
Thread[{x, y, z} == {(2 + Cos[2 s]) Cos[3 s],
(2 + Cos[2 s]) Sin[3 s], Sin[2 s]}]] /.
Thread[{Cos[s], Sin[s]} -> {(1 - u^2)/(1 + u^2), (2 u)/(
1 + u^2)}], {x, y, z}, u] // FullSimplify
```

On the other hand, it does seem that Cox/Little/O’Shea is asking you to do it the long way, so here’s the “painful” route:

```
GroebnerBasis[Append[TrigExpand[
Thread[{x, y, z} ==
{(2 + Cos[2 s]) Cos[3 s], (2 + Cos[2 s]) Sin[3 s],
Sin[2 s]}]] /. Thread[{Cos[s], Sin[s]} -> {a, b}],
a^2 + b^2 == 1], {x, y, z}, {a, b}] // FullSimplify
```

It’s not too hard to do a sanity check of the results of `GroebnerBasis[]`

. Here’s one way (to be done after executing the previous snippet):

```
% /. Thread[{x, y, z} -> {(2 + Cos[2 s]) Cos[3 s],
(2 + Cos[2 s]) Sin[3 s], Sin[2 s]}] // Simplify
```

If everything went well, you should be getting a list containing a bunch of zeroes.

As I am writing this, I don’t have *Mathematica* installed on the computer I’m using. It can happen that one of the the two options I gave might give a longer list of ideals, but it’s guaranteed that one would be a subset of the other.

Yes your expressions are correct.

For further practices, you can try:

$$\begin{align*}

\cos(ax)&=\frac{e^{iax}+e^{-iax}}{2}\\

\sin(ax)&=\frac{e^{iax}-e^{-iax}}{2i}

\end{align*}$$

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