Abstract
This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [<a href="#1">1</a>,<a href="#2">2</a>] for example) requires the use of power series, following the seminal work of Lagrange [<a href="#3">3</a>]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.
Highlights
Trinomial equation has always driven the attention of researchers
Trinomial equations appear in several applications, among which some of the most recent are in financial mathematics [16] and motion analysis of aircraft planar trajectories [17]
Our contribution is based on the work in [18] and [19,20,21], where a theory is developed to treat an algebraic equation of the form: f (y) = φ(y) + x ψ(y) = 0, (1)
Summary
The first contributions beyond the purely algebraic approach dates back to [4,5], went through the works [6,7], to arrive to more recent fundamental contributions [8,9,10,11,12]. To put into practice Equation (4), the sums sr need to be computed according to (2); to this aim, recall that, given an n–th degree polynomial: p(y) = an yn + an−1 yn−1 + . Besso [22] who worked at the quintic equation: y5 + y − x = 0 We will adapt his procedure, considered in [23] for the trinomial of degree n : yn + y − x = 0.
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