Assume you have an ideal $I\subseteq\mathbb{Z}[X_1,\ldots,X_n]$ of the polynomial ring in $n$ variables over the integers. For any field $\Bbbk$, I can consider the ideal $I(\Bbbk):=I\otimes_{\mathbb{Z}}\Bbbk\subseteq\Bbbk[X_1,\ldots,X_n]$. It is well-known that $1\notin I(\mathbb C)$ implies that for prime numbers $p\gg 0$, we also have $1\notin I(\overline{\mathbb F}_p)$ where $\overline{\mathbb F}_p$ denotes an algebraic closure of the field with $p$ Elements.

Geometrically, it means that the scheme $Z(I)$ which is defined over any field has $\overline{\mathbb F}_p$-rational points for large enough $p$ if (and actually, only if) it has $\mathbb{C}$-rational points. Even more geometric, if the system of equations in $I$ has a solution over the complex numbers, then it admits solutions in characteristic $p$ for large $p$.

I am interested in explicit bounds on $p_0\in\mathbb N$ such that the above holds for all prime numbers $p>p_0$. For this purpose, let us assume that $I$ is generated by polynomially many polynomials of degree less or equal to $d$, for some fixed $d$. Then $p_0=p_0(n)$ depends on the number of variables involved.

In the 1978 paper *Bounds on transfer principles for algebraically closed and complete discretely valued fields* by Scott Brown, we get a roughly driple exponential bound:

TheoremIf $S$ is a first-order statement in the language with plus, minus, times, and constant symbols $0, 1, 10, 11, 100,\ldots$ whose representation is $m$ characters long, and $p>2^{2^{2^{3m}}}$, then $S$ holds in the theory of algebraically closed fields of characteristic $p$ if and only if it holds in the theory of algebraically closed fields of characteristic zero.

Observe that since we fixed the degree $d$, the number $m$ from that theorem will be a polynomial in $n$, because for a fixed degree $d$, the number of monomials in $n$ variables of degree $d$ grows polynomially and I assumed that $I$ is generated by polynomially many elements.

**Edit.** I now realize that I also have to assume that the coefficients of my generators are not too large, so let us assume that as well.

**Question.** Is there a reference for a substantially better bound? I have reasons to believe that a doubly exponential bound (with possibly some polynomial in $n$ in the exponent) should be possible.