1.1 A brief introduction to computational algebraic geometry  (Page 6/20)

Thus, since we may take $n$ and $m$ to be relatively prime, as long as $n$ and $m$ aren't both odd (which would lead to a triple with $b$ odd instead) we have that $a=n^2-m^2$ , $b=2nm$ , and $c=n^2+m^2$ . This gives a sort of “parametrization” of the Pythagorean triples and in particular makes it easy to show there are infinitely many of them (and not just by multiplying by a constant).

Exercises

1. Recall that a rational function $x\left(t\right)$ is one of the form $x\left(t\right)=\frac{p\left(t\right)}{q\left(t\right)}$ , where $p$ and $q$ are polynomials. Show that the following curves are rational by finding non-constant functions $x\left(t\right)$ and $y\left(t\right)$ such that $f\left(x\right(t),y(t\left)\right)\equiv 0$ . Then use a computer to graph the curve from the implicit function and then from the parametrization to verify that they coincide (at least for some section of the curve). Hint: Try using a substitution such as $t=\frac{y}{x}$ or $t=\frac{y}{x^2}$ .
   1. $f(x,y)=y^2-x^3$
   2. $f(x,y)=x^2-y^2-(x-2y)(x^2+y^2)$
   3. $f(x,y)=x^5-x{y}^2+y^3$
   4. $f(x,y)=3x-2y-y^2$
   5. $f(x,y)=x^5-x^4+x^{2}y-y^2$
   6. $f(x,y)=x^2+2xy+y^2-y$
   7. $f(x,y)=x^2-2x-y+1$
2. A cardioid is defined by the polar equation $r=1-cos\theta$ . Find an implicit polynomial equation $f(x,y)=0$ for the cardioid, and show that
   $$\left(x,\left(,t,\right),,,y,\left(,t,\right)\right)=\left((,cos,t,),(,1,-,cos,t,),,,(,sin,t,),(,1,-,cos,t,)\right)$$
   is a (non-rational) parametrization of it.
3. Recall the definition of an affine equivalence from last week. Show that affine equivalence preserves rationality. That is, show that if $f(x,y)=g\left(\phi \right(x,y\left)\right)$ for some affine equivalence $\phi$ and $V\left(g\right)=\left\{\right(x,y):f(x,y)=0\}$ is rational then $V\left(f\right)$ is also rational.
4. 1. Show that any nonempty conic is affine equivalent to one with no constant term, i.e. a conic of the form $f(x,y)=ax+by+c{x}^2+dxy+e{y}^2$ .
   2. Let $f(x,y)=(ax+by)+(c{x}^2+dxy+e{y}^2)=f_1+f_2$ be irreducible, where $f_i$ is the purely degree $i$ part of the polynomial. Prove that $V\left(f\right)$ is rational.
   3. Show that any irreducible conic is rational.
   4. Now, let $f(x,y)$ by an irreducible degree $n$ polynomial such that $f=f_{n-1}+f_n$ , so that $f$ has no terms of degree less than $n-1$ . Prove that $f(x,y)=0$ is a rational curve.
5. On the last homework, we began investigating the solution of equations like $n=4x+6y$ and $n=5x+8y$ . We discovered that which numbers are expressible in the form $ax+by$ for $x,y$ integers seems to have a lot to do with the the greatest common divisor of $a$ and $b$ . In fact, it turns out that the standard method of computing the g.c.d. $d$ of $a$ and $b$ can help us solve the equation $d=ax+by$ for integers $x$ and $y$ . This computational method is called “Euclid's algorithm” and works by repeated division with remainder as follows:
   $$\begin{array}{cccc}\hfill b& =q_{1}a\hfill & & +r_1\hfill \\ \hfill a& =q_2{r}_1\hfill & & +r_2\hfill \\ \hfill r_1& =q_3{r}_2\hfill & & +r_3\hfill \\ & \cdots \hfill & & \\ \hfill r_{n-2}& =q_n{r}_{n-1}\hfill & & +r_n\hfill \\ \hfill r_{n-1}& =q_{n+1}{r}_n\hfill & & +0\hfill \end{array}$$
   The algorithm eventually terminates when it gets a zero remainder (since the remainders get smaller at each step). At that point the g.c.d. of $a$ and $b$ is known to be the last non-zero remainder $r_n$ .
   1. Why does Euclid's algorithm work to find the g.c.d.? [Hint: The common divisors of $r_1$ and $r_2$ are the same as the common divisors of $r_2$ and $r_3$ . (Why?)]
   2. How does Euclid's algorithm allow us to write the g.c.d. $r_n$ in the form $ax+by$ ? Use it to solve $68x+173y=1$ .
   3. In the integers modulo 173, what is the multiplicative inverse of 68?

Ideals and monomial orders

Given an algebraic plane curve $f(x,y)=0$ , we've been looking at the problem of finding a rational parametrization $\left(x\right(t),y(t\left)\right)$ for it, where $x\left(t\right)$ and $y\left(t\right)$ are rational functions of $t$ . In several examples (and in a couple of general cases) we've been able to show that curves are rational and exhibit rational parametrizations.