Given the polynomials
$f(x,y)=4x^{12}+7y^{18}-1=0$ and $g(x,y)=5x^{10}+9y^{14}-1=0$
I have to figure out:
a) if the real solutions set of the above system is empty or not
b) if the set of real solutions is finite or infinite and the maximum number of solutions in the set
I don't really know how to to solve the first one.. But I suspect that it should be not empty so that I can go on with the rest of the questions.
a) ?
b) Since the only polynomial that has infinite solutions is $f(x,y)=0$ and the polynomials of the given system are diferent than that, then the system has finite solutions. Also if $n$ is the number of roots of $f(x,y)$ and $m$ is the number of roots of $g(x,y)$ then the solutions of the system is $s=min\left \{ n,m \right \}$
Any ideas on a) ? Also do you think my solution to b) is correct?
Best Answer
Here is an elementary way to show there are no real solutions:
$$5x^{10}+9y^{14}=1 \implies x^{10} \leqslant \tfrac15,\: y^{14} \leqslant \tfrac19$$
$$\implies 4x^{12}+7y^{18}\leqslant \frac4{5^{6/5}}+\frac7{9^{9/7}}<1$$