This answer seems to imply that there is something special about a system of $ n $ quadratic forms $ q_1, \dots, q_n $ in $ n+1 $ unknowns $ x_1, \dots, x_{n+1} $. I want to understand better why this is special.
The general idea seems to be that to the system of quadratic forms $ q_1=0, \dots, q_n=0 $ you add another quadratic form $$ q_{n+1}:=\sum_{i=1}^{n+1} x_i^2 $$ but break homogeneity by instead requiring that $$ q_{n+1}=1 $$ This rules out the zero solution. Then it seems to be the implied that the system of degree 2 equations $ q_1=0, \dots, q_n=0, q_{n+1}=1 $ has a solution in the variables $ x_1, \dots, x_{n+1} $. I think the idea is that this is basically equivalent to finding a solution in $ \mathbb{C}P^n $ or $ \mathbb{R}P^n $ to $ n $ constraints.
Let me ask some, perhaps naive, questions to give a sense of what I'm interested in.
Is it the case that a system of $ n $ independent quadratic forms in $ n+1 $ variables always has a nonzero solution?
Is it the case that a system of $ n $ independent quadratic forms in $ n+1 $ variables always has a nonzero real solution, so long as $ q_1,\dots,q_n $ are independent from $ q_{n+1}=q_{n+1}:=\sum_{i=1}^{n+1} x_i^2 $ ?
And to be clear when I say that the $ q_1 \dots q_n $ are independent I mean that removing any one constraint $ q_i=0 $ will always increase the solution set.