Suppose $p(x , y) = ax^2 + bxy + cy^2$ is a homogenous real polynomial Suppose $p(x , y) = ax^2 + bxy + cy^2$ is a homogenous real polynomial of degree $2$ such that $b^2 < 4ac$, and $q(x , y)$ is a homogenous real polynomial of degree $3$. Show that we can find $k>0$ such that $p(x , y) = q(x , y)$ has no roots inside the disc $x^2 + y^2 <k$, except $(0 , 0)$.

Assume WLOG that $a, c > 0$ and pick $\epsilon > 0$ so small that $(a-\epsilon)(c-\epsilon) > b^2$. Then for any $(x, y) \in \Bbb{R}^2$ we have

$$ p(x, y) \geq \epsilon (x^2 + y^2). $$

(Or, if spectral theory is applicable, pick $\epsilon$ as the smaller of two positive eigenvalues associated to the quadratic form $p(x, y)$.) Now let $M = \sup \\{ |q(x, y)| : x^2 + y^2 = 1 \\}$. Then writing $r = (x^2 + y^2)^{1/2}$, we have

$$ |q(x, y)| = r^3 |q(x/r, y/r)| \leq Mr^3 \leq \frac{Mr}{\epsilon} \cdot \epsilon r^2. $$

Choose $k > 0$ so that $k < \epsilon / M$. Then whenever $0 < r < k$, we have

$$ |q(x, y)| < \epsilon r^2 \leq p(x, y). $$