Notes on polynomials $(1+X{)}^n+(-1{)}^n(X^n+1)$ concerning the regularity problem for symmetric power sums in 3 variables

Abstract

Let $K$ be a field and $f_n(X)=(X+1{)}^n+(-1{)}^n(X^n+1)\in K[X]$, for each $n\in \mathbb{N}$. This note shows that the polynomials $f_m(X)$ and $f_{m^{\prime }}(X)$ are relatively prime, for some distinct indices $m$ and $m^{\prime }$ at most equal to $100$, if and only if the product $m{m}^{\prime }$ is divisible by $6$.

13 pages, no figures, an Appendix is added with results of calculations giving a positive answer to Question (1), for admissible pairs of positive integers at most equal to 605

References in corpus (1)

• Regular sequences of power sums and complete symmetric polynomials