Abstract Generated abstract
The paper studies the problem of extending a polynomial by adding higher degree terms so that all roots of the resulting polynomial lie on the unit circle. Using Newton’s formulas, the problem is reduced to representing the first power sums of inverse roots by points of modulus one, and a constructive successive procedure is given for choosing arguments and sufficiently large multiplicities to satisfy the required equations. The method also indicates how, without loss of generality, a measure zero subset of the unit circle may be avoided by introducing small parameter variations while preserving the prescribed power sums.
Full Text
Mathematics
L. I. Gavrilov
On \(K\)-Extendability of Polynomials
(Presented by Academician P. S. Aleksandrov on 11 VI 1960)
In the present work we consider the problem of extendability of polynomials to the circle \(|z|=1\) of the plane of a complex variable, and give a new, very simple method for solving this problem.
The problem consists in adding to the terms of a given polynomial
\[ f(z)=1+a_1z+a_2z^2+\cdots+a_nz^n \]
new summands in such a way that the roots of the polynomial
\[ f_1(z)=f(z)+a_{n+1}z^{n+1}+\cdots+a_mz^m \]
lie on the circle \(|z|=1\).
From Newton’s formulas
\[ \begin{aligned} s_1+a_1&=0,\\ s_2+a_1s_1+2a_2&=0,\\ &\ldots\ldots\ldots\ldots\\ s_n+a_1s_{n-1}+\cdots+na_n&=0 \end{aligned} \]
we see that \(s_1,s_2,\ldots,s_n\) do not change under extension; moreover, if the roots of the polynomial \(f_1(z)\) are denoted by \(z_1,z_2,\ldots,z_m\), then
\[ s_k=\sum_{j=1}^{m} z_j^{-k},\qquad k=1,2,\ldots,n. \tag{1} \]
It is proved that, whatever \(s_1,s_2,\ldots,s_n\) may be, one can always choose \(z_1,z_2,\ldots,z_m\), equal to unity in modulus, so that equations (1) will be satisfied. Equations (1) admit solutions
\[ s_k=\sum_{j=1}^{n} N_j\bigl(e^{ik\alpha_j}+e^{-ik\alpha_j}\bigr) \bigl(\varepsilon_{1j}^{k}+\varepsilon_{2j}^{k}+\cdots+\varepsilon_{jj}^{k}\bigr)e^{ik\psi_j}, \qquad k=1,2,\ldots,n, \]
where \(\varepsilon_{1j},\varepsilon_{2j},\ldots,\varepsilon_{jj}\) are the roots of the equation
\[ z^j=1, \]
satisfying the conditions
\[ \begin{aligned} \varepsilon_{1j}+\varepsilon_{2j}+\cdots+\varepsilon_{jj}&=0,\\ \varepsilon_{1j}^{2}+\varepsilon_{2j}^{2}+\cdots+\varepsilon_{jj}^{2}&=0,\\ &\ldots\ldots\ldots\ldots\\ \varepsilon_{1j}^{j}+\varepsilon_{2j}^{j}+\cdots+\varepsilon_{jj}^{j}&=j; \end{aligned} \]
\(N_j\) are sufficiently large positive integers denoting the multiplicities of the corresponding roots. Here \(\alpha_j\) and \(\psi_j\) are found successively for \(j=1,2,\ldots,n\).
For \(j=1\), from the first equation we find
\[ s_1=N_1\left(e^{i\alpha_1}+e^{-i\alpha_1}\right)e^{i\psi_1} =2N_1\cos\alpha_1\,e^{i\psi_1}. \]
If \(s_1=\rho_1 e^{i\varphi_1}\), then to determine \(\alpha_1\) and \(\psi_1\) we obtain two equations
\[ \rho_1=2N_1\cos\alpha_1,\qquad e^{i\varphi_1}=e^{i\psi_1}, \]
which can be satisfied by setting
\[ \cos\alpha_1=\frac{\rho_1}{2N_1},\qquad \psi_1=\varphi_1, \]
which is possible for sufficiently large \(N_1\). Having determined \(\alpha_1\) and \(\psi_1\) from the first equation, we next find \(\alpha_2\) and \(\psi_2\) from the second equation. Then we shall have
\[ s_1=2N_1e^{i\psi_1}\cos\alpha_1+2(\varepsilon_{12}+\varepsilon_{22})N_2e^{i\psi_2}\cos\alpha_2, \]
\[ s_2=2N_1e^{2i\psi_1}\cos 2\alpha_1 +2(\varepsilon_{12}^{2}+\varepsilon_{22}^{2})N_2e^{2i\psi_2}\cos 2\alpha_2. \]
Consequently, we shall have
\[ s_2-2N_1e^{2i\psi_1}\cos 2\alpha_1 =4N_2e^{2i\psi_2}\cos 2\alpha_2, \]
and, if
\[ s_2-2N_1e^{2i\psi_1}\cos 2\alpha_1=\rho_2 e^{i\varphi_2}, \]
then we take
\[ 2\psi_2=\varphi_2,\qquad \cos 2\alpha_2=\frac{\rho_2}{4N_2}, \]
which is possible for sufficiently large \(N_2\). In an analogous manner, from the \(k\)-th equation \(\alpha_k\) and \(\psi_k\) will be found after \(\alpha_1,\ldots,\alpha_{k-1}, \psi_1,\ldots,\psi_{k-1}\) have been determined from the preceding equations.
Further, without violating the generality of the theorem, one may remove from the circle \(|z|=1\) a subset of points of measure zero. In this case the following solutions of equations (1) are possible:
\[ s_k=\sum_{j=1}^{n}N_j\left(e^{ik\beta_j}+e^{-ik\beta_j}\right) \left(e^{ik\alpha_j}+e^{-ik\alpha_j}\right) \left(\varepsilon_{1j}^{k}+\varepsilon_{2j}^{k}+\cdots+\varepsilon_{jj}^{k}\right)e^{ik\psi_j}, \]
\[ k=1,2,\ldots,n. \]
Using these formulas we successively compute \(\alpha_1,\ldots,\alpha_k,\psi_1,\ldots,\psi_n\) first for \(\beta_j=0\); then we vary \(\beta_j\) on the segment \([0,\varepsilon]\), ensuring that the roots
\[ \left(e^{i\beta_j}+e^{-i\beta_j}\right) \left(\varepsilon_{1j}^{-1}+\varepsilon_{2j}^{-1}+\cdots+\varepsilon_{jj}^{-1}\right) \left(e^{i\alpha_j}+e^{-i\alpha_j}\right)e^{-i\psi_j} \]
fall into the remaining set. At the same time \(d\alpha_j/d\beta_j=0\) for \(\beta_j=0\). For example, for \(j=1\),
\[ s_1=2N_1e^{i\psi_1}\cos\alpha_1\left(e^{i\beta_1}+e^{-i\beta_1}\right). \]
Then
\[ \frac{s_1}{4N_1\cos\beta_1}=e^{i\psi_1}\cos\alpha_1. \]
Putting \(\beta_1=0\), we find \(\alpha_1\) and \(\psi_1\) by the preceding method. Next we vary \(\beta_1\) on the segment \([0,\varepsilon]\), ensuring that the roots
\[ \left(e^{i\beta_1}+e^{-i\beta_1}\right) \left(e^{i\alpha_1}+e^{-i\alpha_1}\right)e^{-i\psi_1} \]
fall into the remaining set. Since \(s_1=\rho_1 e^{i\varphi_1}\), we have \(\cos\alpha_1=\rho_1/4N_1\cos\beta_1\). Differentiating, we obtain
\[ -\sin\alpha_1\,\frac{d\alpha_1}{d\beta_1} =\frac{\rho_1\sin\beta_1}{4N_1\cos^2\beta_1}, \]
and for \(\beta_1=0\), \(d\alpha_1/d\beta_1=0\), i.e. \(\alpha_1\) is approximately constant when \(\beta_1\) varies on the segment \([0,\varepsilon]\) for sufficiently small \(\varepsilon\).
Received
10 VI 1960
CITED LITERATURE
- L. I. Gavrilov, Izv. Fiz.-matem. obshch. pri Kazansk. gos. univ., 12, ser. 3 (1940).
- P. S. Aleksandrov, A. N. Kolmogorov, Introduction to the Theory of Functions of a Real Variable, Moscow—Leningrad, 1938.