On intersections of a curve with hyperplanes, and the number of zeros of certain functions (Q1977513)

The paper presents the following sufficient criterion that a given curve \((x_1(t),\dots, x_n(t))\in \mathbb{R}^n\) \((a\leq t\leq b)\) has at most \(n\) points (including the multiplicity) with any hyperplane: denoting \(y_{-1}(t)= t\), \(y_1(t)= x_1(t)\) and recursively \(y_k= {d\over dy_{k-1}}\cdots{d\over dy_1} x_k\) for \(k= 2,\dots, n\), then \(dy_j/dy_{j- 1}\neq 0\) (\(a\leq t\leq b\); \(j= 1,\dots,n\)). The familiar Descartes estimate (the number of positive roots of a polynomial does not exceed the number of its terms) appears as a very particular subcase.