# On some classes of irreducible polynomials

## TL;DR

This paper introduces new families of irreducible polynomials, including a generalization for near-separated polynomials and the largest known family of HIP polynomials, expanding understanding in polynomial irreducibility over various fields.

## Contribution

It generalizes previous results by establishing irreducibility conditions for near-separated polynomials and introduces the largest known family of HIP polynomials in multiple variables.

## Key findings

- Near-separated polynomials plus a non-zero constant are always irreducible.
- Established the largest known family of HIP polynomials in several variables.
- Results extend to fields of positive characteristic with modifications.

## Abstract

The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, then $F(x,y)+r$ is always irreducible for any constant $r$ different from zero. We also provide the biggest known family of HIP polynomials in several variables. These are polynomials $p(x_1,\ldots,x_n) \in K[x_1,\ldots,x_n]$ over a zero characteristic field $K$ such that $p(h_1(x_1),\ldots,h_n(x_n))$ is irreducible over $K$ for every $n$-tuple $h_1(x_1),\ldots,h_n(x_n)$ of non constant one variable polynomials over $K$. The results can also be applied to fields of positive characteristic, with some modifications.