Resonance set of a polynomial and problem of formal stability

A. B. Batkhin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2016), pp. 6-24.

Résumé

Let $f_n(x)$ be a monic polynomial of degree $n$ with real coefficients \begin{equation*} f_n(x)\stackrel{def}{=} x^n+a_1x^{n-1}+a_2x^{n-2}+\dotsb+a_n. \end{equation*} The space $\Pi\equiv\mathbb R^n$ of its coefficients $a_1,\dotsc a_n$ is called the coefficient space of $f_n(x)$. A pair of roots $t_i$, $t_j$, $i,j=1,\dotsc,n$, $i\neq j$, of the polynomial $f_n(x)$ is called $p:q$-commensurable if $t_i:t_j=p:q$. Resonance set $\mathcal R_{p:q}(f_n)$ of the polynomial $f_n(x)$ is called the set of all points of $\Pi$ at which $f_n(x)$ has at least a pair of $p:q$-commensurable roots, i.e. \begin{equation*} \mathcal R_{p:q}(f_n)=\{P\in\Pi: \exists\, i,j=1,\dotsc,n, t_i:t_j=p:q\}. \end{equation*} The chain $\mathrm{Ch}_{p:q}^{(k)}(t_i)$ of $p:q$-commensurable roots of length $k$ is called the finite part of geometric progression with common ratio $p/q$ and scale factor $t_i$, each member of which is a root of the polynomial $f_n(x)$. The value $t_i$ is called the generating root of the chain. Any partition $\lambda$ of degree $n$ of $f_n(x)$ defines a certain structure of its $p:q$-commensurable roots and it corresponds to some algebraic variety $\mathcal V_l^i$, $i=1,\dotsc,p_l(n)$ of dimension $l$ in the coefficient space $\Pi$. The number of such varieties of dimension $l$ is equal to $p_l(n)$ and total number of all varieties consisting the resonance set $\mathcal R_{p:q}(f_n)$ is equal to $p(n)-1$. Algorithm for parametric representation of any variety $\mathcal V_l$ from the resonance set $\mathcal R_{p:q}(f_n)$ is based on the following Theorem. Let $\mathcal V_l$, $\dim\mathcal V_l=l$, be a variety on which $f_n(x)$ has $l$ different chains roots and the chain $\mathrm{Ch}_{p:q}^{(m)}(t_1)$ has length $m>1$. Let $\mathbf r_l(t_1,t_2,\dotsc,t_l)$ is a parametrization of variety $\mathcal V_l$. Therefore the following formula \begin{equation*} \mathbf r_l(t_1,\dotsc,t_l,v)=\mathbf r_l(t_1,\dotsc,t_l)+\frac{p(v-p^{m-1}t_1)}{t_1(p^m-q^m)}\left[\mathbf r_l(t_1,\dotsc,t_l)-\mathbf r_l((q/p)t_1,\dotsc,t_l)\right] \end{equation*} gives parametrization of the part of variety $\mathcal V_{l+1}$, on which there exists $\mathrm{Ch}_{p:q}^{(m-1)}(t_1)$, simple root $v$ and other chains of roots are the same as on the initial variety $\mathcal V_l$. From the geometrical point of view the Theorem means that part of variety $\mathcal V_{l+1}$ is formed as ruled surface of dimension $l+1$ by the secant lines, which cross its directrix $\mathcal V_l$ at two points defined by such values of parameters $t_1^1$ and $t_1^2$ that $t_1^1:t_1^2=q:p$. If $f_n(x)$ has on the variety $\mathcal V_{l+1}$ pairs of complex-conjugate roots it is necessary to make continuation of obtained parametrization $\mathbf r_l(t_1,\dotsc,t_l,v)$. Resonance set of a cubic polynomial $f_3(x)$ can be used for solving the problem of formal stability of a stationary point (SP) of a Hamiltonian system with three degrees of freedom. Let Hamiltonian function $H(\mathbf z)$ expand in SP $H(\mathbf z)=\sum_{i=2}^\infty H_i(\mathbf z)$, where $\mathbf z=(\mathbf q,\mathbf p)$, $\mathbf q$ and $\mathbf p$ — are coordinates and momenta, $H_i(\mathbf z)$ — are homogeneous functions of degree $i$. Characteristic polynomial $f(\lambda)$ of the linearized system $\dot{\mathbf z}=JA\mathbf z$, $A=\mathrm{Hess} H_2$, can be considered as a monic cubic polynomial. Resonance sets $\mathcal R_{p:q}(f_n)$ for $p=1,4,9,16$, $q=1$, give the boundaries of subdomains in $\Pi$, where Bruno's Theorem of formal stability [9] can be applied.

Détail

Keywords: elimination theory, subresultant, computer algebra, formal stability of a stationary point, resonance set.

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