# Deterministic Genericity for Polynomial Ideals

## TL;DR

This paper explores various notions of genericity in algebraic geometry and commutative algebra, providing algebraic characterizations and a deterministic algorithm for certain stable positions in characteristic zero.

## Contribution

It introduces a unified framework for stability notions, offers algebraic characterizations, and presents a deterministic algorithm for achieving generic positions in characteristic zero.

## Key findings

- Deterministic algorithm for generic positions in characteristic zero
- Stable positions are characterized algebraically
- Quasi-stability is achievable in any characteristic

## Abstract

We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent algebraic characterisations are provided. It is shown that in characteristic zero the corresponding generic positions can be obtained with a simple deterministic algorithm. In positive characteristic, only adapted stable positions are reachable except for quasi-stability which is obtainable in any characteristic.