Discriminants of special quadrinomials

TL;DR

This paper derives explicit formulas for the discriminants of specific quadrinomials using elementary methods, extending previous work and providing insights into their algebraic properties.

Contribution

It introduces new explicit formulas for discriminants of quadrinomials of the form x^n + a x^k + b x + c, using elementary approaches, expanding on prior matrix-based results.

Findings

Explicit discriminant formulas for x^n + a x^k + b x + c with k in {2,3,n-1}

Notes on discriminants of x^{2n} + a x^n + b x^l + c for n > 2l

Simplified derivation methods compared to previous matrix approaches

Abstract

Finding an effective formula for describing a discriminant of a quadrinomial (a formula which can be easily computed for high values of degrees of quadrinomials) is a difficult problem. In 2018 Otake and Shaska using advanced matrix operations found an explicit expression of $\Delta(x^n+t(x^2+ax+b))$. In this paper we focus on deriving similar results, taking advantage of alternative elementary approach, for quadrinomials of the form $x^n+ax^k+bx+c$, where $ k \in \{2,3,n-1\}$. Moreover, we make some notes about $\Delta(x^{2n}+ax^n+bx^l+c)$ such that $n>2l$.