On some classes of irreducible polynomials

03/20/2019

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by   Jaime Gutierrez, et al.

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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,…,x_n) ∈ K[x_1,…,x_n] over a zero characteristic field K such that p(h_1(x_1),…,h_n(x_n)) is irreducible over K for every n-tuple h_1(x_1),…,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.