Some remarks on the non-real roots of polynomials

Abstract

Let ð–¿ ∈ â„(ð‘¡)[ð‘¥] be given by ð–¿(ð‘¡, ð‘¥) = ð‘¥^ð‘› + ð‘¡ · g(ð‘¥) and β₁ < ··· < β_ð‘š the distinct real roots of the discriminant ∆_{(ð–¿,ð‘¥)}(ð‘¡) of ð–¿(ð‘¡, ð‘¥) with respect to ð‘¥. Let γ be the number of real roots of

                  

For any ξ > |β_m|, if ð‘›âˆ’s is odd then the number of real roots of ð–¿(ξ,ð‘¥) is γ + 1, and if ð‘›âˆ’s is even then the number of real roots of ð–¿(ξ,ð‘¥) is γ, γ + 2 if ts > 0 or t_s < 0 respectively. A special case of the above result is constructing a family of degree 𑛠≥ 3 irreducible polynomials over â„š with many non-real roots and automorphism group S_ð‘›.