On the continued fraction expansion of the unique root in F(p) of the equation x⁴+x²-Tx-1:12=0 and other related hyperquadratic expansions

2010/09/30 by Lasjaunias, Alain
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1009.6139

Abstract

In 1985, Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, he came across a particular equation of degree 4 in characteristic p=13. This equation has an analogue for all primes p>=5. There are two patterns for the continued fraction of the solution of this equation, according to the residue of p modulo 3. We describe this pattern in the first case, considering especially p=7 and p=13. in the second case we only give indications.

Related

• On a quartic equation and two families of hyperquadratic continued fractions in pwer series fields
• On a particular hyperquadratic continued fraction in F(p) with p>2
• On hyperquadratic continued fractions in power series fields over a finite field
• Sur le développement en fraction continue d'une généralisation de la cubique de Baum et Sweet
• Hyperquadratic continued fractions over a finite field of odd characteristic with partial quotients of degree 1