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.

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.