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Numerical results for Waring's problem in $ {\rm GF}[q,\,x]$


Author: William A. Webb
Journal: Math. Comp. 27 (1973), 193-196
MSC: Primary 12C05
DOI: https://doi.org/10.1090/S0025-5718-1973-0325581-X
MathSciNet review: 0325581
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Abstract: Let p be a prime and let K and $ {A_i}$ denote polynomials whose coefficients are elements of the finite field with p elements. Problems concerning the expression of an arbitrary polynomial K as sums of a small number of squares or cubes of polynomials $ {A_i}$ are discussed. In the problems treated the degrees of the $ {A_i}$ are restricted to be as small as possible. In particular, it is shown that at least five cubes are necessary and that three squares seem to suffice in all but one special case.


References [Enhancements On Off] (What's this?)

  • [1] L. Carlitz, "On the representation of a polynomial in a Galois field as the sum of an even number of squares," Trans. Amer. Math. Soc., v. 35, 1933, pp. 397-410. MR 1501692
  • [2] L. Carlitz, "On the representation of a polynomial in a Galois field as the sum of an odd number of squares," Duke Math. J., v. 1, 1935, pp. 298-315. MR 1545879
  • [3] L. Carlitz, "Sums of squares of polynomials," Duke Math. J., v. 3, 1937, pp. 1-7. MR 1545967
  • [4] L. Carlitz, "The singular series for sums of squares of polynomials," Duke Math. J., v. 14, 1947, pp. 1105-1120. MR 9, 337. MR 0023304 (9:337b)
  • [5] Eckford Cohen, "Sums of an even number of squares in $ {\text{GF}}[{p^n},x]$," Duke Math. J., v. 14, 1947, pp. 251-267. MR 9, 81. MR 0021571 (9:81b)
  • [6] Eckford Cohen, "Sums of an even number of squares in $ {\text{GF}}[{p^n},x]$. II," Duke Math. J., v. 14, 1947, pp. 543-557. MR 9, 176. MR 0022233 (9:176b)
  • [7] William Leahey, "Sums of squares of polynomials with coefficients in a finite field," Amer. Math. Monthly, v. 74, 1967, pp. 816-819. MR 0220706 (36:3758)
  • [8] R. E. A. C. Paley, "Theorems on polynomials in a Galois field," Quart. J. Math., v. 4, 1933, pp. 52-63.
  • [9] William Webb, "Waring's problem in $ {\text{GF}}[q,x]$," Acta Arith. (To appear.) MR 0313190 (47:1745)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0325581-X
Keywords: Sums of squares, polynomials over finite fields, Waring's problem
Article copyright: © Copyright 1973 American Mathematical Society

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