Say $X\subset \mathbb{P}^{n}$ be a $\mathbb{Q}$ variety of degree $\beta$.
It is well known and not hard to show that if $X=Z(f_1,...,f_{k})$ where $f\in\mathbb Z[x_0,...,x_n]$ are polynomials of height $h$ and degree $\beta$ then $h(X)\leq poly_{n}(\beta,h)$, where $h(X)$ is the logarithmic height of $X$.
(See for instance http://www.maia.ub.edu/~sombra/talks/mega2003/mega2003.pdf)
It will be useful for me to have an opposite direction. That is, if $h(X)$ is known, to have an estimate for the height of $f_{i}$ that will suffice to form $X$.
Unfortunately I don't really have a good idea. I think the best approach is to define $h'(X)$ to be the minimal height of a system of equations giving $X$ (where the height of the system is the maximum among the height of the equations) and to show that $h,h'$ are equivalent in the sense that :
$h'=\Theta_{n}(h)$
or at the very least $h'\leq poly_{n}(h),\;h\leq poly_{n}(h') $.
Does anyone have an idea or a reference for these kinds of definitions?
Thank you very much!