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Zbl 0758.11042
Habsieger, L.
Représentations des groupes et identités polynomiales. (Representations of groups and polynomial identities). (French)
[J] Sémin. Théor. Nombres Bordx., Sér. II 3, No.1, 1-11 (1991). ISSN 0989-5558

The easier Waring problem is to compute the least number $s = v(k)$ such that every integer is a sum or difference of $s$ integral $k$-th powers. The only values $v(k)$ known precisely are $v$(1) =1 and $v$(2) = 3. The best upper bounds for $v(k)$ known for small $k$ are obtained using polynomial identities of the following form: a sum or difference of $n \ k$-th powers of polynomials in $x$ with integral coefficients equals a polynomial of degree one. For example, the identity $(x + 1)\sp 2-x\sp 2=2x$ with $k$ = 2 and $n$ = 2 shows that $v(2) \le 3$. R. Norrie (1911) found the identity $(ax + b\sp 3)\sp 4 -(ax -b\sp 3)\sp 4 + (-bx + a\sp 3)\sp 4 -(bx + a\sp 3)\sp 4 = 8ab(b\sp 8-a\sp 8)x$ with $k$ = 4, $n$ = 4, containing two parameters. Rao (1938) found a similar identity with $k$ = 6, $n$ = 8. Vaserstein (1987) found a similar identity with $k$ = 8, $n$ = 12. This identity contains three parameters. The author makes a change of parameters which brings Vaserstein's identity to a more symmetric form. He applies group representations to obtain polynomial identities including Norrie's, Rao's, and Vaserstein's identities. However he did not obtain any new bounds for $v(k)$.
[L.N.Vaserstein and E.R.Wheland (University Park)]

MSC 2000:: *11P05 Waring's problem and variants

Keywords: easier Waring problem; sum or difference of higher powers; polynomial identities

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