Let $R=\Bbb R[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I want to compare minimum degrees of $f_1$ and $f_2$ with minimum degrees of $g_1$ and $g_2$. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

I am most interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.