The paper under review is concentrated on two issues:
(1) presentation of a multivariate polynomial over a field \(K\) as a sum of univariate polynomials in linear forms defined over \(K\). More precisely, let \(d\) be a positive integer and \(K\) a field of characteristic not dividing \(d\). If the characteristic is equal to \(0\) or \(>d\) it is known [see A. Schinzel, J. Théor. Nombres Bordx. 14, No. 2, 647–666 (2002; Zbl 1067.11012)] that every polynomial \(F \in K[x_1, \dots, x_n]\) of degree \(d\) can be written as \(F=\sum_{i=1}^{m}f_{i}(\ell_{i})\), where \(m \leq \binom{n+d-1}{d}\),\(f_{i} \in K[t]\) (univariate polynomial) and \(\ell_{i}\in K[x_1, \ldots, x_n]\) is a linear form (\(1 \leq m \leq m\)). For \(d \leq 3\) there is a better bound: \(m \leq \binom{n+d-2}{d-1}\) [see A. Schinzel, Collq. Math. 92, 67–79 (2002; Zbl 0987.12001)]; the authors conjecture that this holds in general and they prove (by Theorem 1) this conjecture for fields of big enough cardinality (in particular for infinite fields).
(2) presentation of a form, in particular a zero form, as the sum of powers of linear forms projectively distinct defined over an algebraically closed field.
Moreover, an upper bound on the number of summands in presentations of all (not only generic) polynomials and forms of a given number of variables (i.e., \(n\)) and degree (i.e., \(d\)) is given. Also some special cases of these problems are studied and other nice intermediate results are given.
Added in 2012: The correction text states: In Lemma 2.3 (p. 211) replace “If” by “If \(S(k,d,s)\) is finite and”.