Polynomial Distribution and Sequences of Irreducible Polynomials over Finite Fields, Original Research Article

Letk=GF(q) be the finite field of orderq. Letf1(x),f2(x)set membership, variantk[x] be monic relatively prime polynomials satisfyingn=deg f1>deg f2greater-or-equal, slanted0 andf1(x)/f2(x)≠g1(xp)/g2(xp) for anyg1(x),g2(x)set membership, variantk[x]. WriteQ(x)=f1(x)+tf2(x) and letKbe the splitting field ofQ(x) overk(t). LetGbe the Galois group ofKoverk(t).Gcan be regarded as a subgroup ofSn. For any cycle patternλofSn, letπλ(f1, f2, q) be the number of square-free polynomials of the formf1(x)−αf2(x) (αset membership, variantk) with factor patternλ(corresponding in the natural way to cycle pattern). We give general and precise bounds forπλ(f1, f2, q), thus providing an explicit version of the estimates for the distribution of polynomials with prescribed factorisation established by S. D. Cohen in 1970. For an application of this result, we show that, ifqgreater-or-equal, slanted4, there is a (finite or infinite) sequencea0,a1, …set membership, variantk, whose length exceeds 0.5 log q/log log q, such that for eachngreater-or-equal, slanted1, the polynomialfn(x)=a0+a1x+…+anxnset membership, variantk[x] is an irreducible polynomial of degreen. This resolves in one direction a problem of Mullen and Shparlinski that is an analogue of an unanswered number-theoretical question of A. van der Poorten.