Ratio vectors of polynomial-like functions

Let $p(x)$ be a polynomial like function of the form $p(x)=(x-r_1{)}^{m_1}\mathrm{.}\mathrm{.}\mathrm{.}(x-r_N{)}^{m_N}$, where $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$ are given positive real numbers and $r_1<r_2<\mathrm{.}\mathrm{.}\mathrm{.}<r_N$. Let $\{x_k\}$ be the critical points of $p$ in $(r_k,r_{k+1})$ and define the ratios ${\sigma }_k={\displaystyle \frac{x_k-r_k}{r_{k+1}-r_k}},k=1,2,\mathrm{.}\mathrm{.}\mathrm{.},N-1$. $({\sigma }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\sigma }_{N-1})$ is called the \QTR{it}{ratio vector} of $p$. We extend some some of the results on ratio vectors from earlier papers for the case when $m_1=\mathrm{.}\mathrm{.}\mathrm{.}=m_N=1$, that is, for polynomials of degree $n$ with $n$ distinct real roots. For N=3, we find necessary and sufficient conditions for $({\sigma }_1,{\sigma }_2)$ to be a ratio vector. We also simplify as well as extend some of the proofs for N=4. In particular we show that $\dfrac{m_{k}}{m_{k}+… +m_{N}}<\sigma_{k}<\dfrac{m_{1}+… +m_{k}}{m_{1}+… +m_{k+1}}$, and that the monotonicity of the ratios does not hold in general for $N\ge 3$. For N=3 we find necessary and sufficient conditions on $m_1,m_2,m_3$ which imply that ${\sigma }_1<{\sigma }_2$. We also prove some results for general $N$ using the theory of \QTR{it}{Groebner bases} and projective elimination theory. One consequence is that for any $N\geq 2,\sigma_{1},…,\sigma_{N-1}$satisfyanontrivialpolynomialequationin$N-1$ variables with real coefficients.