The number of centered forms for a polynomial
If f: \(R^ n\to R\) and \(c\in R^ n\) are given, every representation of f as expression \(f(x)=f(c)+g(x,c)\cdot (x-c)\) is called a centered form of f at c. In this definition, g: \(R^{2n}\to R\) is called the kernel of the centered form and the dot denotes the standard inner product in \(R^ n\). The concept of centered forms is due to \textit{R. E. Moore} [Interval analysis (1966; Zbl 0176.133)]. The interval extension of a centered form over an n-dimensional interval X is denoted by \(f(X)=f(c)+g(X,c)\cdot (X-c).\) Interval extensions of centered forms are an excellent means to approximate the range of f over X from without, i.e., f(X)\(\supseteq \{f(x):\) \(x\in X\}\). The kernel g underlying the centered form idea is not uniquely defined and, furthermore, different kernels will, in general, provide different inclusions f(X) of the range. The search for kernels which lead to optimum inclusions of the range is an old and interesting problem. In connection which such considerations the authors develop formulas for the number of different centered forms of a given multi-variate polynomial at a point c. Methodically, the number of different centered forms of such a polynomial is reduced to the number of different arrangements of multi-variate variables. Conversely, an algorithm is presented that generates a centered form at c if an arrangement of the variables is given. The algorithm generates different centered forms when the arrangements are different. Hence it suffices to develop recursive formulas for the number of different arrangements.
scientific article; zbMATH DE number 4080769