Numerical results for Waring's problem in

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 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 are discussed. In the problems treated the degrees of the 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.

**[1]**Leonard Carlitz,*On the representation of a polynomial in a Galois field as the sum of an even number of squares*, Trans. Amer. Math. Soc.**35**(1933), no. 2, 397–410. MR**1501692**, https://doi.org/10.1090/S0002-9947-1933-1501692-9**[2]**Leonard Carlitz,*On the representation of a polynomial in a Galois field as the sum of an odd number of squares*, Duke Math. J.**1**(1935), no. 3, 298–315. MR**1545879**, https://doi.org/10.1215/S0012-7094-35-00121-1**[3]**Leonard Carlitz,*Sums of squares of polynomials*, Duke Math. J.**3**(1937), no. 1, 1–7. MR**1545967**, https://doi.org/10.1215/S0012-7094-37-00301-6**[4]**L. Carlitz,*The singular series for sums of squares of polynomials*, Duke Math. J.**14**(1947), 1105–1120. MR**0023304****[5]**Eckford Cohen,*Sums of an even number of squares in 𝐺𝐹[𝑝ⁿ,𝑥]*, Duke Math. J.**14**(1947), 251–267. MR**0021571****[6]**Eckford Cohen,*Sums of an even number of squares in 𝐺𝐹[𝑝ⁿ,𝑥]. II*, Duke Math. J.**14**(1947), 543–557. MR**0022233****[7]**William Leahey,*Sums of squares of polynomials with coefficients in a finite field*, Amer. Math. Monthly**74**(1967), 816–819. MR**0220706**, https://doi.org/10.2307/2315800**[8]**R. E. A. C. Paley, "Theorems on polynomials in a Galois field,"*Quart. J. Math.*, v. 4, 1933, pp. 52-63.**[9]**William A. Webb,*Waring’s problem in 𝐺𝐹[𝑞,𝑥]*, Acta Arith.**22**(1973), 207–220. MR**0313190**

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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