Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $$n$$ many degree $$2$$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $$\mathbb Z[x_1,\dots,x_n]$$ with absolute value of coefficients bound by $$2^{m}$$ what is the maximum number of bounded primitive (gcd of coordinates $$1$$) integer points possible?

1. Is it $$d^{O(mn)}$$ for general degree $$d$$ systems or much lower/higher?

2. Is there any sharp bound at all?

1. Even for the simple equation $$x_1x_2=n$$ where $$n$$ is a product of $$O\big(\frac{\log n}{\log\log n}\big)$$ distinct primes we can expect exponential number of primitive solutions $$(x_1,x_2)\in\mathbb Z^2$$. However here we have homogeneous system.