Biro 1994 problem turan concerning sums powers complex

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lemma_1: Proves that each new polynomial coefficient either makes a specified Newton--Girard expression large or forces quantitative growth of the consecutive coefficient partial sum.

theorem_1: Derives two power-sum identities from Newton--Girard and combines them with the coefficient-sum dichotomy to prove a strict one-half bound.

theorem_2: Biró's refinement of his one-half bound to systems whose first m members equal one: the largest modulus among the first n-m+1 power sums exceeds m times 1/2 + (1/8)(m/n) + (3/64)(m/n)^2, which for m = 1 sharpens Theorem 1 by a term of order 1/n.

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András Biró, On a problem of Turán concerning sums of powers of complex numbers. Acta Mathematica Hungarica 65 (1994), no. 3, 209--216. DOI: 10.1007/BF01875148.

The copy read for this card is a 448-page scan of the published journal volume. This article occupies physical PDF pp. 223--230, corresponding to printed pp. 209--216. Its title, author, volume, issue, year, and printed page range are visible in the scan. The source record is https://real-j.mtak.hu/7463/. The article begins at physical p. 223 of that scan. The scan's volume front matter (p. 2) prints "Copyright (c) 1994 by Akadémiai Kiadó, Budapest." and, at its foot, "Printed in Hungary", and its back matter (p. 447) states that "copyright will be vested in the publisher"; the notice is the volume's and covers the article, every other right reserved.

Power-sum quantity

For complex numbers $z_1,\ldots,z_n$, write

$$S_j=\sum_{t=1}^n z_t^j.$$

The paper studies

$$R_n= \min_{\substack{z_1,\ldots,z_n\in\mathbb C\\ \max_{1\leq t\leq n}|z_t|=1}} \ \max_{1\leq j\leq n}|S_j|.$$

The minimum may equivalently be normalized by requiring $z_1=1$: a number of maximum modulus can be relabeled first and rotated to $1$, while a tuple already having $z_1=1$ can first be divided by a member of maximum modulus, which does not increase any $|S_j|$.

Reconstructed result

[[analysis/biro_1994_problem_turan_concerning_sums_powers_complex/theorem_1|Theorem 1]] proves that, for arbitrary complex $z_1,\ldots,z_n$ with $z_1=1$,

$$\max_{1\leq j\leq n}|S_j|>\frac12.$$

The proof forms the polynomial with roots $z_2,\ldots,z_n$, applies two precisely stated Newton--Girard identities, and invokes the fully reconstructed [[analysis/biro_1994_problem_turan_concerning_sums_powers_complex/lemma_1|Lemma 1]]. That lemma gives a dichotomy between a large Newton--Girard right-hand side and growth of consecutive coefficient partial sums. The theorem handles both the persistent-growth case and the first-failure case, then takes $\alpha=\pi/4$.

This gives the exact existential conclusion in Problem 519 with $c=1/2$.

At publication, Biró presented this as an improvement on Atkinson's lower estimates. The article cites the 1961 bound $R_n>1/6$, whose source is filed at [[analysis/atkinson_1961_sums_powers_complex_numbers/_index|Atkinson (1961)]], and it records the then-known upper estimates of Komlós, Sárközy, and Szemerédi. These are historical statements from the 1994 article rather than claims about the present best bounds.

Separate repeated-one refinement

[[analysis/biro_1994_problem_turan_concerning_sums_powers_complex/theorem_2|Theorem 2]], printed on p. 212 and proved on pp. 212--215, states that if $m\geq1$, $n>m$, and

$$z_1=\cdots=z_m=1,$$

then

$$\max_{1\leq j\leq n-m+1}|S_j| > m\left(\frac12+\frac18\frac mn+ \frac3{64}\left(\frac mn\right)^2\right).$$

Its proof uses the separate Lemmas 2 and 3. Remark 1 on printed p. 215 says that optimizing the angular parameter improves the coefficient of $(m/n)^2$ from $3/64$ to $1/16$. Remark 2 on p. 215 reads off from the proof a condition on the first $n-m$ power sums that forces $|S_{n-m+1}|>m\cos\alpha$, and Remark 3 on pp. 215--216 outlines, for $m=1$, the bound $R_n>\frac12+\frac{0.159}n$ for sufficiently large $n$, without computing the constant. Those arguments were read for scope and statement fidelity, but they are not reconstructed or assigned complete-proof credit here; the theorem page records the statement, the remarks and a proof pointer.

Stated precisely, Lemma 2 says that for $z\neq0$, $0<\alpha<\pi/2$, and $A>0$, at least one of

$$|1-Az|^2\geq\sin^2\alpha \left(1+\frac{\cos^2\alpha}{A+\sin^2\alpha}\right) \tag{5}$$

and

$$|1+z|\geq1+\cos\alpha|z| \tag{6}$$

holds. For Lemma 3, define

$$\prod_{t=m+1}^n(x-z_t) =x^{n-m}+b_1x^{n-m-1}+\cdots+b_{n-m}.$$

For $0<\alpha<\pi/2$ and each $1\leq k\leq n-m$, that lemma gives at least one of

$$\begin{aligned} \left|m(1+b_1+\cdots+b_{k-1})-kb_k\right|^2 \geq{}&m^2\sin^2\alpha \left(1+\frac{\cos^2\alpha}{n/m-\cos^2\alpha}\right)\\ &\mathrel{}\cdot|1+b_1+\cdots+b_{k-1}|^2, \end{aligned} \tag{10}$$

or

$$|1+b_1+\cdots+b_k| \geq|1+b_1+\cdots+b_{k-1}|+\cos\alpha|b_k|. \tag{11}$$

If (11) holds through $k=s\leq n-m$, its iterated conclusion is

$$|1+b_1+\cdots+b_s| >\cos\alpha(1+|b_1|+\cdots+|b_s|).$$

These are statement transcriptions from printed pp. 212--213, except that the numerator $\cos^2\alpha$ in (10) is printed as $\cos^2$; the proof on p. 214 uses $\cos^2\alpha$. Their proofs and their use in Theorem 2 remain outside the selected complete chain.

Later history and scope

The 1994 constant is historical. Biró's later An improved estimate in a power sum problem of Turán, Indagationes Mathematicae 11 (2000), no. 3, 343--358, DOI 10.1016/S0019-3577(00)80003-8, proves on printed p. 344 that there is an effectively computable absolute $q>1/2$ such that $R_n>q$ for every $n$; the article does not compute a concrete value of $q$.

A different 2000 article, An upper estimate in Turán's pure power sum problem, Indagationes Mathematicae 11 (2000), no. 4, 499--508, DOI 10.1016/S0019-3577(00)80018-X, proves $\limsup_{n\to\infty}R_n<1$. It obtains $R_n<5/6$ for all sufficiently large $n$, and its addendum records Harcos's computation $\limsup_{n\to\infty}R_n<0.69368$. These later papers were checked at their published statement pages to delimit the 1994 result; their proofs are not part of this reconstruction. The checked copies were author-hosted renderings of the published journal articles, not editions of the 1994 article. No optimality or current-best claim is made.

All eight pages of the 1994 article were visually inspected. The complete proof compiled here is the Theorem 1--Lemma 1 chain on printed pp. 210--211, together with the definitions on p. 209. Newton--Girard is the only external algebraic identity; its exact interface is stated on the theorem page. The planar geometry is proved locally. No formal proof build was run.

Bears on. Problem 519: the problem asks for an absolute $c>0$ with $\max_{1\leq k\leq n}|\sum_iz_i^k|>c$ whenever $z_1=1$; Theorem 1 proves this with $c=1/2$, and the case $m=1$ of Theorem 2 gives the bound $\frac12+\frac1{8n}+\frac3{64n^2}$ for each $n\geq2$, which does not yield an absolute constant above $1/2$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.