# Bad Primes in Computational Algebraic Geometry

## TL;DR

This paper presents a technique for rational reconstruction in computational algebraic geometry that ensures correct results even with the presence of bad primes, by relying on a sufficiently large set of good primes.

## Contribution

It introduces a new rational reconstruction method that tolerates bad primes, enhancing the robustness of computations over the rationals in algebraic geometry.

## Key findings

- The method guarantees correct results with enough good primes.
- Implemented examples demonstrate practical effectiveness.
- Applicable in various computational algebraic geometry problems.

## Abstract

Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. In this note, we describe a technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the selected set of primes is large enough. We give a number of illustrating examples which are implemented using the computer algebra system Singular and the programming language Julia. We discuss applications of our technique in computational algebraic geometry.