Sums of fourth powers of polynomials over a finite field of characteristic 3.(English)Zbl 1213.11195

Let $$F$$ be a finite field of characteristic 3 with $$q$$ elements. A sum $M = M_{1}^4+\ldots+ M_{s}^4$ of biquadrates of polynomials $$M_1,\dots, M_{s}\in F[T]$$ is a strict one if $$4\deg M_i < 4 + \deg M$$ for each $$i= 1,\ldots, s.$$ Here are the main results. Let $$P\in F[T]$$ of degree $$\geq 329.$$ If $$q>81$$ is congruent to $$1$$ (mod $$4$$), then $$P$$ is the strict sum of $$9$$ biquadrates; if $$q=81$$ or $$q>3$$ is congruent to $$3$$ (mod $$4$$), then $$P$$ is the strict sum of $$10$$ biquadrates. If $$q=3,$$ every $$P\in F[T]$$ which is a sum of biquadrates is a strict sum of $$12$$ biquadrates, if $$q=9,$$ every $$P\in F[T]$$ which is a sum of biquadrates and whose degree is not divisible by $$4$$ is a strict sum of $$8$$ biquadrates; every $$P\in F[T]$$ which is a sum of biquadrates, whose degree is divisible by $$4$$ and whose leading coefficient is a biquadrate is a strict sum of $$7$$ biquadrates.

MSC:

 11T55 Arithmetic theory of polynomial rings over finite fields 11P05 Waring’s problem and variants
Full Text:

References:

 [1] R. Balasubramanian, J.-M. Deshouillers, F. Dress, Problème de Waring pour les bicarrés. I: schéma de la solution , C.R. Acad. Sci., Paris, Sér. I 303 (1986), 85-88. · Zbl 0594.10039 [2] N. Bourbaki, Eléments de mathématique, Fascicule XI, chap. 5, Hermann, Paris (3rd. Ed), 1973. [3] M. Car, New bounds on some parameters in the Waring problem for polynomials over a finite field , Contemporary Math, 461 , (2008) 59-77. · Zbl 1220.11149 [4] M. Car, L. Gallardo, Sums of cubes of polynomials , Acta Arith. 112 (2004), 41-50 · Zbl 1062.11078 [5] M. Car, L. Gallardo, Waring’s problem for biquadrates over a finite field of odd characteristic , Funct. Approx. Comment. Math., 37 .1 (2007), 39-50. · Zbl 1144.11084 [6] H. Davenport, On Waring’s problem for fourth powers , Annals of Math. (2) 40, (1939), 731-747 JSTOR: · Zbl 0024.01402 [7] J.-M. Deshouillers; K. Kawada and T. Wooley, On sums of sixteen biquadrates , Mem. Soc. Math. Fr (N.S) n$$\,^\circ$$ 100 (2005). · Zbl 1080.11066 [8] L. Gallardo, On the restricted Waring problem over $$\F_2^n[t]$$, Acta Arith. 42 (2000), 109-113. · Zbl 0948.11034 [9] C. Small, Sums of powers in large finite fields , Proc. Amer. Math. Soc. 65 (1977), 35-36. · Zbl 0328.12016 [10] M. R. Stein, Surjective stability in dimension $$0$$ for $$K_2$$ and related functors , Trans. Amer. Math. Soc. 178 (1973), 165-191. · Zbl 0267.18015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.