Nice-polynomials

A. È. Sergeev ; V. O. Bushueva

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2023), pp. 83-91

Résumé

Decomposition of polynomials into multipliers and finding their roots is one of the main problems of algebra. For an arbitrary polynomial of arbitrary degree, it is quite difficult to find all the roots. In the present paper nice-polynomials, i.e., polynomials whose all roots are integers, but the roots of the derivative is also integers. It is known how to obtain and construct a general nice polynomial of the second, third and fourth degrees over the field of rational numbers $Q$. The problem of obtaining and constructing even concrete (not general) nice polynomials is not simple, since in general this problem is reduced to solving general Diophantine equations, the solution of which, most often, is impossible (in integers). Therefore, for degree 5 and higher, in general, very cumbersome Diophantine equations are obtained, in which even partial solutions are not easy to obtain and the question of constructing a general nice-polynomial remains open until now. In this paper, we provide particular examples of constructing nice polynomials of 5th degree. Specific examples are given for each case. This method allows finding private nice polynomials of higher degree. Has been hypothesized that for any natural $n$, a larger that one, there are nice-polynomials of the degree n. At the same time, the question of finding polynomials of arbitrary degree in which the roots of the polynomial itself, its derivative and second derivative would be different integers remains unresolved.

Détail

Mots-clés : polynomial, decomposition of a polynomial, nice-polynomial.
Keywords: the root of polynomial