Primes of bad reduction for systems of polynomial equations

Abstract

Consider polynomials $F_1,\dots,F_s$ in $\K[X_1,\dots,X_n]$ over a field $\K$, their zero-set $V(F_1,\dots,F_n)$ in $\Kbar^n$ and its decomposition into equidimensional components $V_0,\dots,V_n$ (with $V_i$ either empty or of dimension $i$ for all $i$). To each $V_i$, we can associate its Chow forms, which are polynomials in new variables $(U_{k,j})_{0\le k\le i, 0 \le j \le n}$, uniquely defined up to a scalar factor. These Chow forms completely characterize $V_i$: we can recover equations for $V_i$ from them, and their degree is $(i+1)$ times the degree of $V_i$. We discuss the situation when the $F_i$'s have integer coefficients, and study the question of when the Chow forms of the $V_i$'s defined as above can be reduced modulo $p$ to give Chow forms of the equidimensional components of $V(F_1 \bmod p,\dots,F_s \bmod p)$. We show that this is the case as soon as $p$ does not divide a certain nonzero integer $\Delta$ of height $O(n^{14} s h d^{3n+4})$, with $d$ and $h$ bounds on respectively the degrees and heights of the $F_i$'s.