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Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.

Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.

Is it possible to find a polynomial $\tilde q$ such that $\tilde q\geqslant0$, $\deg \tilde q\leqslant 2$, and $\tilde q|_S=q|_S$?

Comments

• If the system contains only one polynomial, then the answer is yes.

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.

Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.

Is it possible to find a polynomial $\tilde q$ such that $\tilde q\geqslant0$, $\deg \tilde q\leqslant 2$, and $\tilde q\ |_S=q|_S$?

Comments

• If the system contains only one polynomial, then the answer is yes.

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.

Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.

Is it possible to find a nonnegative quadratic polynomial $\tilde q$ such that $\tilde q|_S=q|_S$?

Comments

• If the system contains only one polynomial, then the answer is yes.

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.

Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.

Is it possible to find a polynomial $\tilde q$ such that $\tilde q\geqslant0$, $\deg \tilde q\leqslant 2$, and $\tilde q|_S=q|_S$?

Comments

• If the system contains only one polynomial, then the answer is yes.