## 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
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### References:

  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  N. Bourbaki, Eléments de mathématique, Fascicule XI, chap. 5, Hermann, Paris (3rd. Ed), 1973.  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  M. Car, L. Gallardo, Sums of cubes of polynomials , Acta Arith. 112 (2004), 41-50 · Zbl 1062.11078  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  H. Davenport, On Waring’s problem for fourth powers , Annals of Math. (2) 40, (1939), 731-747 JSTOR: · Zbl 0024.01402  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  L. Gallardo, On the restricted Waring problem over $$\F_2^n[t]$$, Acta Arith. 42 (2000), 109-113. · Zbl 0948.11034  C. Small, Sums of powers in large finite fields , Proc. Amer. Math. Soc. 65 (1977), 35-36. · Zbl 0328.12016  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
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