EVERY POLYNOMIAL OVER A FIELD CONTAINING 𝔽16 IS A STRICT SUM OF FOUR CUBES AND ONE EXPRESSION A2 + A

Title & Authors
EVERY POLYNOMIAL OVER A FIELD CONTAINING 𝔽16 IS A STRICT SUM OF FOUR CUBES AND ONE EXPRESSION A2 + A
Gallardo, Luis H.;

Abstract
Let q be a power of 16. Every polynomial $\small{P\in\mathbb{F}_q}$[t] is a strict sum $\small{P=A^2+A+B^3+C^3+D^3+E^3}$. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial $\small{Q\in\mathbb{F}_q}$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $\small{Q=F^2+F+tG^2}$. This improves for such q's and such Q's a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F,G,H for the strict representation $\small{Q=F^2}$+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.
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English
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