Page 1. International Electronic Journal of Algebra Volume 9 (2011) 85-102 ON SPLITTING PERFECT P... more Page 1. International Electronic Journal of Algebra Volume 9 (2011) 85-102 ON SPLITTING PERFECT POLYNOMIALS OVER Fpp Luis H. Gallardo and Olivier Rahavandrainy Received: 10 March 2010; Revised: 29 July 2010 Communicated by Abdullah Harmancı Abstract. ...
Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixe... more Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixed sum of 11 biquadrates and it is also a strict sum of 16 biquadrates. For the strict mixed sum representation, a supplementary biquadrate is required for q ∈ {13, 17, 25, 29} and four supplementary biquadrates for q = 5. Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {7, 13} is a strict sum of 7 cubes. One supplementary cube is required for q = 13 and two for q = 7.
Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixe... more Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixed sum of 11 biquadrates and it is also a strict sum of 16 biquadrates. For the strict mixed sum representation, a supplementary biquadrate is required for q ∈ {13, 17, 25, 29} and four supplementary biquadrates for q = 5. Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {7, 13} is a strict sum of 7 cubes. One supplementary cube is required for q = 13 and two for q = 7.
There is no odd unitary multiperfect number. When the quotient σ (n)/n = 2 then n is called perfe... more There is no odd unitary multiperfect number. When the quotient σ (n)/n = 2 then n is called perfect and analogously when the quotient σ * (n)/n = 2 then n is called unitary perfect. So, trivially, all unitary perfect numbers are even (see Theorem 3 for an analogue for polynomials).
We prove among several results that under mild conditions any polynomial in F q [t] is a strict s... more We prove among several results that under mild conditions any polynomial in F q [t] is a strict sum of k 4 kth powers improving on an exponential (k 2 2 k+1 ) bound of Car-Effinger-Hayes.
Page 1. International Electronic Journal of Algebra Volume 9 (2011) 85-102 ON SPLITTING PERFECT P... more Page 1. International Electronic Journal of Algebra Volume 9 (2011) 85-102 ON SPLITTING PERFECT POLYNOMIALS OVER Fpp Luis H. Gallardo and Olivier Rahavandrainy Received: 10 March 2010; Revised: 29 July 2010 Communicated by Abdullah Harmancı Abstract. ...
Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixe... more Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixed sum of 11 biquadrates and it is also a strict sum of 16 biquadrates. For the strict mixed sum representation, a supplementary biquadrate is required for q ∈ {13, 17, 25, 29} and four supplementary biquadrates for q = 5. Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {7, 13} is a strict sum of 7 cubes. One supplementary cube is required for q = 13 and two for q = 7.
Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixe... more Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {5, 13, 17, 25, 29}, is a strict mixed sum of 11 biquadrates and it is also a strict sum of 16 biquadrates. For the strict mixed sum representation, a supplementary biquadrate is required for q ∈ {13, 17, 25, 29} and four supplementary biquadrates for q = 5. Every polynomial P ∈ F q [t] where gcd (q, 6) = 1 and q / ∈ {7, 13} is a strict sum of 7 cubes. One supplementary cube is required for q = 13 and two for q = 7.
There is no odd unitary multiperfect number. When the quotient σ (n)/n = 2 then n is called perfe... more There is no odd unitary multiperfect number. When the quotient σ (n)/n = 2 then n is called perfect and analogously when the quotient σ * (n)/n = 2 then n is called unitary perfect. So, trivially, all unitary perfect numbers are even (see Theorem 3 for an analogue for polynomials).
We prove among several results that under mild conditions any polynomial in F q [t] is a strict s... more We prove among several results that under mild conditions any polynomial in F q [t] is a strict sum of k 4 kth powers improving on an exponential (k 2 2 k+1 ) bound of Car-Effinger-Hayes.
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