A generalization of Descartes rule of signs and fundamental theorem of algebra (Q648307)

In this paper there are described axiomatically families of real functions defined on some interval and having the property that each linear combination of such functions has less roots than the number of nonzero coefficients (the number of sign changes in the coefficient sequence, respectively). These axioms are satisfied by monomials with real exponents on each interval not containing \(0\), by exponential functions on \(\mathbb{R}\), but not by logarithm functions with different bases. A noticeable feature of the approach is that it requires only elementary tools from mathematical analysis. Equally simple is the proof of the fact that, for \(n\) pairwise distinct functions \(f_j\) from such a family and \(n\) distinct real numbers \(t_i\), the vectors \(\bigl( f_j(t_1), f_j(t_2),\ldots,f_j(t_n) \bigr)^\top\) (\(1\leq j\leq n\)) form a basis of \(\mathbb{R}^n\).