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

1. L. I. Gavrilov, Izv. Fiz.-matem. obshch. pri Kazansk. gos. univ., 12, ser. 3 (1940).
2. P. S. Aleksandrov, A. N. Kolmogorov, Introduction to the Theory of Functions of a Real Variable, Moscow—Leningrad, 1938.