MATHEMATICS

A. N. KOSTOVSKII

DETERMINATION OF STRICT INEQUALITIES BETWEEN THE MODULI OF ROOTS IN THE PROCESS OF TRANSFORMING ALGEBRAIC EQUATIONS BY THE LOBACHEVSKY–GRAEFFE METHOD

(Presented by Academician S. L. Sobolev on 28 V 1962)

Let a polynomial with complex coefficients be given,

\[ f(z)=\sum_{i=0}^{n} a_i z^i,\qquad a_0\ne 0, \tag{1} \]

whose roots are arranged in increasing order of their moduli

\[ 0<|z_1|\leqslant |z_2|\leqslant \cdots \leqslant |z_n|. \tag{2} \]

Construct the polynomial

\[ f_k(z)=\prod_{j=0}^{k-1} f\left(\sqrt[k]{z}\,e^{\frac{2\pi}{k}\sqrt{-1}}\right) =a_0^{(k)}+a_1^{(k)}z+\cdots+a_n^{(k)}z^n=0, \tag{3} \]

whose roots are equal to the \(k\)-th powers of the roots of the original polynomial. The coefficients of the functions (3) are transformed according to the formulas proposed in \((^1,^2)\).

In the present note the following theorem will be proved:

Theorem. For the inequality \(|z_m|<|z_{m+1}|\) between the moduli of the roots of polynomial (1) to hold, it is necessary and sufficient that, for every pair of values \(\chi\) and \(\nu\),

\[ \lim_{k\to\infty} \frac{\left|a_{m-\chi}^{(k)}\right|^{\nu}\left|a_{m+\nu}^{(k)}\right|^{\chi}} {\left|a_m^{(k)}\right|^{\chi+\nu}} =0, \tag{4} \]

where \(1\leqslant \chi\leqslant m,\; 1\leqslant \nu\leqslant n-m\).

Conditions (4) are necessary and sufficient. Only the necessary conditions were indicated by Pólya \((^3)\). Sufficient conditions were found by Valiron \((^4)\), and then refined and extended by Ostrowski to a broader class of functions \((^5)\). In practice, the application of these criteria is connected with the cumbersome construction of special majorizing polynomials for each of the transformed polynomials (3), the so-called Newton diagrams.

Proof. Let

\[ |z_m|<|z_{m+1}|; \tag{5} \]

then

\[ \frac{\left(a_{m-\chi}^{(k)}\right)^{\nu}\left(a_{m+\nu}^{(k)}\right)^{\chi}} {\left(a_m^{(k)}\right)^{\chi+\nu}} = \frac{\left[(-1)^{m-\chi}S\left(z_1^{-k}\ldots z_{m-\chi}^{-k}\right)\right]^{\nu} \left[(-1)^{m+\nu}S\left(z_1^{-k}\ldots z_{m+\nu}^{-k}\right)\right]^{\chi}} {\left[(-1)^m S\left(z_1^{-k}\ldots z_m^{-k}\right)\right]^{\chi+\nu}} = \]

\[ = \left[ \chi!\nu! \sum_{\alpha_1+\alpha_2+\cdots+\alpha_r=\nu,\ \beta_1+\beta_2+\cdots+\beta_p=\chi} \frac{1}{\alpha_1!\cdots\alpha_r!\beta_1!\cdots\beta_p!} \times \right. \]

\[ \left. \times \left( \frac{ (z_1\cdots z_{m-\chi})^{\alpha_1}\cdots (z_{n-m+\chi+1}\cdots z_n)^{\alpha_r} (z_1\cdots z_{m+\nu})^{\beta_1}\cdots (z_{n-m-\nu+1}\cdots z_n)^{\beta_p} } {(z_1\cdots z_m)^{\chi+\nu}} \right)^{-k} \right] \times \]

\[ \times \left[ (\chi+\nu)! \sum_{\lambda_1+\lambda_2+\cdots+\lambda_q=\chi+\nu} \frac{1}{\lambda_1!\cdots\lambda_q!} \left( \frac{ (z_1\cdots z_m)^{\lambda_1}\cdots (z_{n-m+1}\cdots z_n)^{\lambda_q} } {(z_1\cdots z_m)^{\chi+\nu}} \right)^{-k} \right]^{-1} = \frac{\varepsilon(k)}{1+\varepsilon_1(k)}, \tag{6} \]

where \(r=C_n^{m-\varkappa};\ p=C_n^{m+\nu};\ q=C_n^m\); by the symbol \(S\) are denoted elementary symmetric functions of the reciprocals of the roots of the \(k\)-th transformed polynomial (3). The unit in the denominator is obtained when \(\lambda_1=\varkappa+\nu,\ \lambda_2=\cdots=\lambda_q=0\).

The number of terms of which the numerator \(\varepsilon(k)\) in (6) consists does not exceed \((C_n^{m-\varkappa})^\nu (C_n^{m+\nu})^\varkappa\) and does not depend on \(k\). The number of terms \(\varepsilon_1(k)\) is no greater than \((C_n^m)^{\varkappa+\nu}-1\), which also does not depend on \(k\). The term of largest absolute value in \(\varepsilon(k)\) will occur for the values \(\alpha_1=\nu,\ \alpha_2=\cdots=\alpha_r=0,\ \beta_1=\varkappa,\ \beta_2=\cdots=\beta_p=0\):

\[ \left| \frac{(z_1\ldots z_{m-\varkappa})^{-k\nu}(z_1\ldots z_{m+\nu})^{-k\varkappa}} {(z_1\ldots z_m)^{-k(\varkappa+\nu)}} \right| = \left| \frac{(z_{m-\varkappa-1}\ldots z_m)^{k\nu}} {(z_{m+1}\ldots z_{m+\nu})^{k\varkappa}} \right| \leqslant \left| \left(\frac{z_m}{z_{m+1}}\right)^{k\varkappa\nu} \right|. \]

An analogous assertion is valid for the moduli of each of the terms in \(\varepsilon_1(k)\). Consequently, taking into account inequalities (2), (5), for sufficiently large values of \(k\) we may write

\[ \left| \frac{(a_{m-\varkappa}^{(k)})^\nu (a_{m+\nu}^{(k)})^\varkappa} {(a_m^{(k)})^{\varkappa+\nu}} \right| = \frac{ O\left(\left|\dfrac{z_m}{z_{m+1}}\right|^k\right) }{ 1+ O_1\left(\left|\dfrac{z_m}{z_{m+1}}\right|^k\right) }. \]

Hence,

\[ \lim_{k\to\infty} \frac{(a_{m-\varkappa}^{(k)})^\nu (a_{m+\nu}^{(k)})^\varkappa} {(a_m^{(k)})^{\varkappa+\nu}} =0. \]

This implies (4).

Suppose now that, for the coefficient \(a_m^{(k)}\), the equalities (4) hold. The maximum value of \(\varkappa+\nu\) is not greater than \(n\); therefore, for any pair of values \(\varkappa\) and \(\nu\), \(1\leq \varkappa\leq m,\ 1\leq \nu\leq n-m\), it follows that
\[ \varkappa\nu \leq \left(\frac{\varkappa+\nu}{2}\right)^2 \leq \left(\frac{n}{2}\right)^2. \]
Put \(\varepsilon=9^{-n^2/4}\) and choose such an \(h\) that, for every pair of values \(\varkappa,\nu\), the inequalities

\[ \left| \frac{(a_{m-\varkappa}^{(k)})^\nu (a_{m+\nu}^{(k)})^\varkappa} {(a_m^{(k)})^{\varkappa+\nu}} \right| <9^{-n^2/4}, \]

or

\[ \left| \frac{(a_{m-\varkappa}^{(k)})^\nu (a_{m+\nu}^{(k)})^\varkappa} {(a_m^{(k)})^{\varkappa+\nu}} \right|^{1/\varkappa\nu} = \frac{ \left|\dfrac{a_{m-\varkappa}^{(k)}}{a_m^{(k)}}\right|^{1/\varkappa} }{ \left|\dfrac{a_m^{(k)}}{a_{m+\nu}^{(k)}}\right|^{1/\nu} } <9. \]

The last inequality is valid for every pair of values \(\varkappa\) and \(\nu\), hence:

\[ \frac{ \displaystyle \min_{1\leq \nu\leq n-m} \left| \frac{a_m^{(k)}}{a_{m+\nu}^{(k)}} \right|^{1/\nu} }{ \displaystyle \max_{1\leq \varkappa\leq m} \left| \frac{a_{m-\varkappa}^{(k)}}{a_m^{(k)}} \right|^{1/\varkappa} } >9. \]

According to the results of Valiron \((^4)\), from the last inequality follows the validity of the inequality \(|z_m^k|<|z_{m+1}^k|\), and hence also (5).

From the theorem we obtain

Corollary 1. If (5) is satisfied for the polynomial (1), then, beginning with some value \(h_0\), the index \(m\) becomes the principal index of the Newton diagram \(\mathfrak M_{f_k(z)}\) of the function \(f_k(z)\) for \(h<h_0\).

Corollary 2. If the transformation of the polynomials (3) is carried out by the method of “squaring the roots” \((k=2^h)\), then, for \(|z_m|<|z_{m+1}|\), it follows from (4), in particular, that for sufficiently large \(k\)

\[ |a_m^{(k)}|^2 \geq 2\,|a_{m-\nu}^{(k)}a_{m+\nu}^{(k)}|,\quad \text{where }\chi=\nu=1,\ldots,\min(m,n-m). \]

Example

\[ f(z)=1-z+z^2-z^3+\ldots+(-1)^{2n}z^{2n}. \]

Here \(f_k(z)=1-z+z^2-z^3+\ldots+(-1)^{2n}z^{2n}\), \(k=2^1,2^2,\ldots,2^h\). For each coefficient \(a_m^{(k)}\), for each pair of values \(\chi\) and \(\nu\), the left-hand side of equality (4) is equal to one; consequently,

\[ |z_1|=|z_2|=\ldots=|z_n|, \]

although for the coefficients with even indices the generally accepted rule for determining “correctly” changing coefficients is satisfied, i.e.

\[ a_m^{(k+1)}=(a_m^{(k)})^2. \]

Lviv State University
named after Ivan Franko

Received
3 X 1961

REFERENCES

\(^1\) A. N. Kostovskii, Proceedings of the Scientific-Technical Conference: New Developments in Computational Mathematics and Computer Technology, Kiev, 1960.
\(^2\) A. N. Kostovskii, Zhurn. vychislitel’n. matem. i matem. fiz., 1, No. 2, 345 (1961).
\(^3\) G. Pólya, Zs. Math. Phys., 63, 1–2, 275 (1914).
\(^4\) G. Valiron, Sur les fonctions entières d’ordre nul et d’ordre fini et en particulier les fonctions à correspondance régulière, Thèse de Paris, 1914.
\(^5\) A. Ostrowski, Acta Math., 72, 99 (1940).