Lets say one is given a set of $m$ real polynomial equations in $n$ variables, $P_1 = P_2 = P_2 .. = P_m =0$. I understand that there is some theorem which says that if there is no solution to these simultaneous equations then one necessarily will be able to find polynomials $Q_1,Q_2,..,Q_m, R_1, R_2,..,R_k$ such that $deg(P_iQ_i)$ and $deg(R_i^2)$ are all $\leq d$ for some $k$ and $d$ such that, $\sum_{i=1}^m Q_iP_i + \sum_{j=1}^k R_j^2 = -1$
- Can someone kindly reference this theorem and its proof?
Now lets say one is looking at a $d$ which is not sufficiently large. Then one will be looking at the set of all polynomials of the form $S = \sum_{i=1}^m Q_iP_i + \sum_{j=1}^k R_j^2$ such that $deg(P_iQ_i)$ and $deg(R_i^2)$ are all $\leq d$ for various $k$. But this is a convex set in the set of polynomials? And $-1 \not \in S$. Then there will exist a separating hyperplane between S and $-1$ and the normal to that plane gives a degree-d pseudo-distribution on S.
Does someone have a concrete example of the above phenomenon?
Why is this pseudo-distribution indistinguishable from an actual distribution on $S$ if one is only looking at moments $\leq d$?