Let $V$ be a real affine algebraic subset of a finite-dimensional real affine $n$-space defined by the vanishing of finitely many homogeneous polynomials in $n$ real variables $x_1,\ldots,x_n$. Let $f$ and $g$ be two real (nonzero) homogeneous polynomials in $x_1,\ldots,x_n$ of the same degree. Let $H_f$ and $H_g$ be the two real hypersurfaces of affine $n$-space defined by the vanishing of $f$ and $g$ respectively. What are necessary and sufficient conditions on $f$ and $g$ so that $H_f \cap V = H_g \cap V$? This part is I am sure known, and I will look it up shortly. It probably follows from the positivstellensatz.
My second question is this. Suppose we add an extra hypothesis. Let us assume that $H_f \cap V \supseteq H_g \cap V$. What are necessary and sufficient conditions on $f$ and $g$ so that $H_f \cap V = H_g \cap V$? I think this part is also known in real algebraic geometry, but I am posting it because it is a little puzzling for me.
Indeed, I am faced with a situation such as the setup of the second question, and I suspect that $H_f \cap V = H_g \cap V$ in my case. I just don't know how to prove it.
There is the "sum of squares" effect that is true in real algebraic geometry, which makes it much different from the complex case. What I mean is for example this:
Let $f(x,y) = x^2 + y^2$. Let $L_1(x,y) = ax + by$ and $L_2(x,y) = cx + dy$ be two linearly independent forms over $\mathbb{R}$. Let
$$g(x,y) = L_1^2 + L_2^2.$$
Despite $f$ and $g$ be two homogenous polynomials of the same degree which are not multiple of each other in general, yet their real vanishing loci are the same (just the origin). I will refresh my memory on the postivstellensatz in the mean time, but if someone has some comments, or would like to post an answer, then I would greatly appreciate it.
Upon emailing an expert, I now realize that there is a real nullstellensatz, and that the relevant notion to answer both questions is to use the notion of a real radical, see for instance: https://en.wikipedia.org/wiki/Real_radical.