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∈C is a free parameter. It is well-known that, for degree n≥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 [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. 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.

Trinomial equation: the Hypergeometric way.

Ritelli Daniele
Primo
;
Spaletta Giulia
Secondo
2021

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∈C is a free parameter. It is well-known that, for degree n≥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 [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. 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.
Ritelli Daniele; Spaletta Giulia
File in questo prodotto:
File Dimensione Formato  
trinomial-equation-the-hypergeometric-way.pdf

accesso aperto

Tipo: Versione (PDF) editoriale
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione (CCBY)
Dimensione 467.99 kB
Formato Adobe PDF
467.99 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/828621
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact