Some new polynomial discriminant formulas
In the article under review, the authors determine the discriminant of any polynomial, in the variable \(x\), of the form \[ x^{n}+a(x^{n-1}+bx^{n-2}+b^{2}x^{n-3}+\cdot \cdot \cdot +b^{n-2}x+b^{n-1}), \] where \(a\), \(b\), and \(n\geq 3\) are integers such that \(a\) is squarefree and \( \gcd (a,b)=1\), or of the form \[ x^{n}+a((bx)^{n-1}+(bx)^{n-2}+(bx)^{n-3}+\cdot \cdot \cdot +bx+1), \] where \(a\neq 0\), \(b\neq 0\), and \(n\geq 3\) are integers such that \((a,b)\neq (1,1)\) when \(n+1\) is composite and \((a,b)\neq (-1,-1)\) when \(n\) is odd. By generalizing the discriminant formulas, obtained by the second author in [Bull. Aust. Math. Soc. 100, No. 2, 239--244 (2019; Zbl 1461.11138); Acta Arith. 197, No. 2, 213--219 (2021; Zbl 1465.11204)], they also compute the discriminant of any irreducible element of the class \[ x^{n}+a(bx^{k}+c)^{m}, \] where \(n\geq 3\), \(m\geq 1\), \(k\geq 1\), \(a\neq 0\), \(b\), and \(c\neq 0\) are integers such that \(k\) is a proper divisor of \(n\) and \(km<n\), or the class \[ x^{n-km}(x^{k}+a)^{m}+b, \] where \(n\geq 3\), \(m\geq 1\), \(k\geq 1\), \(a\), and \(b\neq 0\) are integers such that \(k\) is a proper divisor of \(n\) and \(km<n\).
scientific article; zbMATH DE number 7441580
scientific article