Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, given $P$, can you find $x_1,\dots,x_n\in\mathbb{R}$ such that $p_i(x_1,\dots,x_n)=0$ for all $i$? I only need to find one solution.

I know that in the general case (arbitrary degree polynomials) this can't be done efficiently. However, I'm wondering if there are any good techniques in the quadratic case, specifically if there is a polynomial time solution.

what?? Also, what are the coefficients of the polynomials in $P$? $\endgroup$