Sums Of Squares On A Lemniscate

Let $S$ be the real variety defined by the polynomial $f=(x^2+y^2)^2-x^2+y^2$:
$$S=\{(x,y) \in \mathbb{R}^2 \,\, |\,\, f(x,y)=0\}.$$

(a) Find the minimal $c \in \mathbb{R}$ such that the polynomial $c-y$ is nonnegative on $S$.

(b) For the value of $c$ from part (a), find a sums of squares certificate for nonnegativity of $c-y$ on $S$.


Hints:

The certificate for part (b) should involve sums of squares of quadratic polynomials and a constant multiple of $f$.


Solution:

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