Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let q be a power of 16. Every polynomial $P\in\mathbb{F}_q$[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 $Q\in\mathbb{F}_q$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $Q=F^2+F+tG^2$. 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 $Q=F^2$+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.

Keywords

References

  1. E. Artin, Geometric Algebra, Wiley (Interscience), 1957
  2. L. H. Gallardo, On the restricted Waring problem over $F_2n$[t], Acta Arith. 92 (2000), no. 2, 109–113
  3. L. H. Gallardo, Waring's problem for cubes and squares over a finite field of even characteristic, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 349–362
  4. L. H. Gallardo, Every strict sum of cubes in $F_4$ [t] is a strict sum of 6 cubes, Port. Math. 65 (2008), no. 2, 227–236
  5. L. H. Gallardo and D. R. Heath-Brown, Every sum of cubes in $F_2$ [t] is a strict sum of 6 cubes, Finite Fields Appl. 13 (2007), no. 4, 981–987 https://doi.org/10.1016/j.ffa.2007.04.001
  6. L. H. Gallardo, O. Rahavandrainy, and L. Vaserstein, Representations of polynomials over finite fields of characteristic two as $A^2+A+BC+D^3$, Finite Fields Appl. 13 (2007), no. 3, 648–658 https://doi.org/10.1016/j.ffa.2005.11.007
  7. R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983
  8. L. N. Vaserstein, Sums of cubes in polynomial rings, Math. Comp. 56 (1991), no. 193, 349–357
  9. L. N. Vaserstein, Ramsey's theorem and Waring's problem for algebras over fields, The arithmetic of function fields (Columbus, OH, 1991), 435–441, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992