Some remarks on the non-real roots of polynomials
- Shuichi Otake shuichi.otake.8655@gmail.com
- Tony Shaska shaska@oakland.edu
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DOI:
https://doi.org/10.4067/S0719-06462018000200067Abstract
Let 𖿠∈ â„(ð‘¡)[ð‘¥] be given by ð–¿(ð‘¡, ð‘¥) = ð‘¥ð‘› + ð‘¡ · g(ð‘¥) and β1 < ··· < β𑚠the 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 > |βm|, 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 ts < 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 Sð‘›.