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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?

Updates

  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.

  2. Related link but irrelevant (here I seek maximum number of integer roots) Real points of zero-dimensional real algebraic varieties.

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