On some properties of Hamel bases connected with the continuity of polynomial functions (Q798538)

The following notations are used: H is a Hamel basis of \(R^ N\), \((H^ i)^*=\{(h_ 1,...,h_ i)| h_ j\in H\), \(h_ j\neq h_ k\), for \(j\neq k\}\). Keeping a single \((h_ 1,...,h_ i)\) for all \((h_{\sigma (1)},...,h_{\sigma (i)})\), where \(\sigma\) is a permutation of 1,...,i, one obtains a set \(\omega_ i((H^ i)^*/\sim_ i)\). Given i,p,\(q\in N\), 1\(\leq i\leq p\leq q\), let \(p_{(i)}\) denote the numbers of all different expansions of p into a sum of i positive integers, expansions differing by succession being considered different, and set \(s_{(i)}=i_{(i)}+(i+1)_{(i)}+...+q_{(i)}\). Given numbers \(\lambda_{i,m,n}\in Q\backslash\{0\}\), \(i=1,...,q\), \(m=1,...,s_{(i)}\), \(n=1,...,i\), for which certain determinants of order \(s_{(i)}\) are different from zero, the following set is defined [cf. \textit{R. Ger}, Ann. Pol. Math. 25, 195-203 (1971; Zbl 0227.39003)] \[ Z(q)=\cup^{q}_{i=1}\cup^{s_{(i)}}_{m=1}\{\lambda_{i,m,1}h_ 1+...+\lambda_{i,m,i}h_ i\in R^ N:\quad (h_ 1,...,h_ i)\in\omega_ i((H^ i)^*/\sim_ i)\}. \] A function \(f_ p:R^ N\to R\) is called monomial of p-th degree, if it is a diagonalization of a p-additive and symmetric function \(F:R^{pN}\to R\), while a function \(f:R^ N\to R\) is called polynomial of q-th order if it admits a representation \(f=f_ 0+f_ 1+...+f_ q\), with \(f_ p\) monomial of pth degree. Generalizing results of \textit{F. B. Jones} [Measure and other properties of a Hamel basis, Bull. Am. Math. Soc. 48, 472-481 (1942)] and \textit{Z. Kominek} [On the continuity of Q-convex functions and additive functions, Aequationes Math. 23, 146-150 (1981)] concerning additive functions, the author proves: if \(T\subset R^ N\) is an analytic set containing a set Z(q), then any polynomial function \(f:R^ N\to R\) of q- th order is continuous whenever 1. f is bounded on T, or 2. the restriction \(f|_ T\) is continuous.