Every polynomial over a field containing F_16 is a strict sum of four cubes and one expression A^2+A
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Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum 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 Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation 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.
Abstract
Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum 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 Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation 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.
참고문헌 (9)
- E.Artin, "Geometric Algebra" (1957), Wiley (Interscience)
- "Acta Arith", 2000, 109~113
- L. H. Gallardo, "Waring’s problem for cubes and squares over a finite field of even characteristic", Bull. Belg. Math. Soc. Simon Stevin, 12(3), pp.349~362 (2005)
- is a strict sum of 6 cubes, "Port. Math", 2008, 227~236
- is a strict sum of 6 cubes, "Fields Appl", 2007, 981~987
- L. H. Gallardo, "Representations of polynomials over finite fields of characteristic two as A2 + A + BC + D3", Finite Fields Appl, 13(3), pp.648~658 (2007)
- R. Lidl, "Finite Fields, Encyclopedia of Mathematics and its Applications", pp.20~ (1983), Addison-Wesley Publishing Company, Advanced Book Program, Reading
- L. N. Vaserstein, "Sums of cubes in polynomial rings", Math. Comp, 56(193), pp.349~357 (1991)
- L. N. Vaserstein, "Ramsey’s theorem and Waring’s problem for algebras over fields, The arithmetic of function fields", pp.435~441 (1992), Ohio State Univ. Math. Res. Inst. Publ