Vaserstein, L. N. Waring’s problem for algebras over fields. (English) Zbl 0624.10049 J. Number Theory 26, 286-298 (1987). For a ring A let \(w_ k(A)\) denote the smallest s such that every sum of kth powers in A is a sum of s kth powers; let \(v_ k(A)\) denote the corresponding infimum for the sum-or-difference of kth powers. This paper gives a detailed study of \(v_ k(A)\) and \(w_ k(A)\) when A is an algebra over a field F. The main result can be stated as follows: \(v_ k(A)\leq k^ 3\) when A is commutative and either F is infinite or A has transcendence degree 1 over F. This statement swallows up several more precise results, which give sharper upper bounds in many situations of interest. Reviewer: C.Small Cited in 1 ReviewCited in 13 Documents MSC: 11C99 Polynomials and matrices 13A99 General commutative ring theory 11P05 Waring’s problem and variants Keywords:Waring’s problem for rings PDFBibTeX XMLCite \textit{L. N. Vaserstein}, J. Number Theory 26, 286--298 (1987; Zbl 0624.10049) Full Text: DOI References: [1] Balasubramanian, R.; Mozzochi, C. J., An improved upper bound for \(G(k)\) in Waring’s problem for relatively small \(k\), Acta Arith., 43, 283-285 (1983/1984) · Zbl 0491.10036 [2] Banerjee, D. P., On the solution of the “easier” Waring problem, Bull. Calcutta Math. Soc., 34, 197-199 (1962) · Zbl 0063.00190 [3] Barrodale, I., A note on equal sums of like powers, Math. Comput., 20, 318-322 (1966) · Zbl 0141.04206 [4] Car, M., Sommes de carres dans \(F_q[X]\), Dissertationes Math., 215 (1983) · Zbl 0531.12016 [5] Choi, M. D.; Dai, Z. D.; Lam, T. Y.; Reznick, B., The pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math., 336, 45-82 (1982) · Zbl 0499.12018 [6] Ellison, W. J., Waring’s problem, Amer. Math. Monthly, 78, No. 1, 10-36 (1971) · Zbl 0205.35001 [7] Fuchs, W. H.J.; Wright, E. M., The ‘easier’ Waring problem, Quart. J. Math. (Oxford), 10, 190-209 (1939) · Zbl 0022.11501 [8] Hua, L. K., On Tarry’s problem, Quart. J. Math. (Oxford), 9, 315-320 (1938) · Zbl 0020.00501 [9] Hunter, W., The representation of numbers by sums of fourth powers, J. London Math. Soc., 16, 177-179 (1941) · Zbl 0028.34606 [10] Joly, J.-R., Sommes de puissances \(d\)-ièmes dans un anneau commutatif, Acta Arith., 17, 37-113 (1970) · Zbl 0206.34001 [11] Karatsuba, A. A., On the function \(G(n)\) in Waring’s problem, Izv. Akad. Nauk Ser. Mat., 49, No. 5, 935-947 (1985), [Russian] · Zbl 0594.10041 [12] Lander, L. J., Three thirteens, Math. Comput., 27, 397 (1973) · Zbl 0279.10018 [13] Mahrota, S. N., On sums of powers, Math. Student, 37, 204-205 (1969) · Zbl 0208.30801 [14] Newman, D. J.; Slater, M., Waring’s problem for the ring of polynomials, J. Number Theory, 11, 477-487 (1979) · Zbl 0407.10039 [15] Rai, T., Easier Waring problem, J. Sci. Res. Benares Hindu Univ., 1, 5-19 (1951) [16] Tornheim, L., Sums of \(n\) th powers in fields of prime characteristic, Duke Math. J., 4, 359-362 (1938) · JFM 64.0093.02 [17] L. N. Vaserstein; L. N. Vaserstein · Zbl 0634.10042 [18] Weil, A., Number of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55, 497-508 (1949) · Zbl 0032.39402 [19] Wright, E. M., An easier Waring’s problem, J. London Math. Soc., 9, 267-272 (1934) · Zbl 0010.10306 [20] Wright, E. M., The Tarry-Escott and the “easier” Waring problems, J. Reine Angew. Math., 311-312, 170-173 (1979) · Zbl 0409.10009 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.