Primes of bad reduction for systems of polynomial equations

TL;DR

This paper investigates the reduction modulo primes of Chow forms associated with algebraic varieties defined by polynomial systems, identifying conditions under which these forms preserve their structure in characteristic p.

Contribution

It introduces criteria for when Chow forms of polynomial system solutions can be reduced modulo p without losing information, based on a specific integer  that depends on the system's parameters.

Findings

Chow forms characterize equidimensional components of polynomial solution sets.

Reduction modulo p preserves Chow forms if p does not divide a certain integer .

The integer 's size depends polynomially on degrees, heights, and number of variables.

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.