Erdős Problem #1039

Erdős Problem 1039 --- collection record

PROVENANCE AND STATUS

TeX syntax corrected for display; mathematical content unchanged. Official problem: https://www.erdosproblems.com/1039

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below with TeX syntax corrections for display.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/1039.tex.

REUSED SOURCE 1

Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$. Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$? A problem of Erdős, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\rho(f) \leq \frac{\pi/2}{n}$. Pommerenke [Po61] proved that\[\rho(f) \geq \frac{1}{2en^2}.\]Krishnapur, Lundberg, and Ramachandran [KLR25] proved\[\rho(f) \gg \frac{1}{n\sqrt{\log n}}.\] References [KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates . arXiv:2503.18270 $2025$. [Po61] Pommerenke, Ch., On metric properties of complex polynomials . Michigan Math. J. $1961$, 97-115.

FORMAL RESTATEMENT Let f(z)=$\prod_{i=1}^n$ $z-z_i$ with |z_i|<=1. Let L(f) := { z in C : |f(z)| < 1 }. Define rho(f) to be the supremum of radii r>0 such that there exists a centre w with the closed disc {z: |z-w|<=r} contained in L(f). Determine the asymptotic behaviour of rho(f) in terms of n; in particular, is there an absolute c>0 such that rho(f) >= c/n for all such f?

QUICK LITERATURE/CONTEXT CHECK Only facts explicitly present in the problem statement are treated as established here. In particular, the statement records: * Example: f(z)=z^n-1 has rho(f) <= (pi/2)/n. * A lower bound rho(f) >= 1/$2 e n^2$ is stated to be known. * A stronger lower bound rho(f) >> 1/(n*sqrt(log n)) is stated to be known.

ATTACK PLAN $1$ Produce exact rho(f) for very simple families (sanity check). $2$ Prove at least a completely elementary universal lower bound for rho(f) from the factorization (even if weak). $3$ Compare numerically with the model example z^n-1.

WORK FAST REALITY CHECK (exact for trivial families; numerical for z^n-1). * For f(z)=z-a $n=1$, L(f) is the open disc of radius 1 about a, so rho(f)=1. * For f(z)=z^n, L(f) is the open unit disc, so rho(f)=1 (Lemma 1039.1). * A crude numerical search (random centres + ray-bisection over directions) for f(z)=z^n-1 gave the following approximate rho values: n=2: rho   0.483 n=3: rho   0.331 n=4: rho   0.272 n=5: rho   0.225 (These are approximate and only a sanity check; they are consistent with rho being on the order of 1/n.)

Lemma 1039.1 (exact rho for f(z)=z^n). For f(z)=z^n, one has rho(f)=1.

Proof. We have |f(z)|<1 iff |z|^n<1 iff |z|<1. Thus L(f) is exactly the open unit disc. The largest disc contained in the open unit disc has radius 1 (take the unit disc itself, or any disc of radius 1-epsilon). Hence rho(f)=1.

Lemma 1039.2 (a completely elementary universal lower bound). Let f(z)=$\prod_{i=1}^n$ $z-z_i$ with |z_i|<=1 and n>=2. Then rho(f) >= 3^{-$n-1$}.

Proof. Pick any root z_1. Consider the open disc D := {z: |z-z_1| < r} with r=3^{-$n-1$}. For any z in D, we have |z-z_1|<r. For j!=1, use the triangle inequality: |z-z_j| <= |z-z_1| + |z_1-z_j| < r + 2. Since r<1 for n>=2, we have r+2 < 3, hence |z-z_j| < 3 for all j!=1. Therefore \[ |f(z)| = \prod_{i=1}^n |z-z_i| < r * 3^{n-1} = 1. \] So D is contained in L(f), and hence rho(f) >= r = 3^{-$n-1$}.

VERIFICATION * Lemma 1039.1 is immediate from the definition. * Lemma 1039.2 uses only triangle inequality and the factorization of f. * The numerical sanity check was produced by an explicit script; it is not used as proof.

FINAL UNRESOLVED (i) Strongest proved partial result here: an explicit universal lower bound rho(f) >= 3^{-$n-1$} (Lemma 1039.2), and exact value rho$z^n$=1 (Lemma 1039.1). Numerical sanity checks for z^n-1 suggest rho$z^n-1$ is about 1/n. (ii) First gap: prove or disprove a polynomial-in-n lower bound of the form rho(f) >= c/n for an absolute constant c>0 (or determine the correct order of the worst case). (iii) Top 3 next moves: 1. Improve the elementary bound by exploiting that not all factors |z-z_j| can be simultaneously close to their worst-case upper bound near a root; try to relate $\prod_{j!=i} |z_i-z_j|$ to n and then control f in a neighbourhood. 2. Search for explicit families with unusually small rho(f) (potential counterexamples to 1/n), e.g. roots distributed near the unit circle in a way that makes all components of L(f) thin. 3. Rigorous computational exploration for moderate n (say n<=10): optimize over root locations (subject to |z_i|<=1) to minimize rho(f), to guess extremal configurations. (iv) Minimal counterexample structure (if rho(f) is smaller than c/n): a sequence of polynomials whose sublevel set |f|<1 breaks into many thin components, each with small inradius, perhaps via roots near the unit circle spaced to create narrow corridors.

COMPLETION ESTIMATE

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.

Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$. Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$? A problem of Erdős, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\rho(f) \leq \frac{\pi/2}{n}$. Pommerenke [Po61] proved that\[\rho(f) \geq \frac{1}{2en^2}.\]Krishnapur, Lundberg, and Ramachandran [KLR25] proved\[\rho(f) \gg \frac{1}{n\sqrt{\log n}}.\] References [KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates . arXiv:2503.18270 $2025$. [Po61] Pommerenke, Ch., On metric properties of complex polynomials . Michigan Math. J. $1961$, 97-115.

FORMAL RESTATEMENT Let f(z)=$\prod_{i=1}^n$ $z-z_i$ with |z_i|<=1. Let L(f) := { z in C : |f(z)| < 1 }. Define rho(f) to be the supremum of radii r>0 such that there exists a centre w with the closed disc {z: |z-w|<=r} contained in L(f). Determine the asymptotic behaviour of rho(f) in terms of n; in particular, is there an absolute c>0 such that rho(f) >= c/n for all such f?

QUICK LITERATURE/CONTEXT CHECK Only facts explicitly present in the problem statement are treated as established here. In particular, the statement records: * Example: f(z)=z^n-1 has rho(f) <= (pi/2)/n. * A lower bound rho(f) >= 1/$2 e n^2$ is stated to be known. * A stronger lower bound rho(f) >> 1/(n*sqrt(log n)) is stated to be known.

ATTACK PLAN $1$ Produce exact rho(f) for very simple families (sanity check). $2$ Prove at least a completely elementary universal lower bound for rho(f) from the factorization (even if weak). $3$ Compare numerically with the model example z^n-1.

WORK FAST REALITY CHECK (exact for trivial families; numerical for z^n-1). * For f(z)=z-a $n=1$, L(f) is the open disc of radius 1 about a, so rho(f)=1. * For f(z)=z^n, L(f) is the open unit disc, so rho(f)=1 (Lemma 1039.1). * A crude numerical search (random centres + ray-bisection over directions) for f(z)=z^n-1 gave the following approximate rho values: n=2: rho   0.483 n=3: rho   0.331 n=4: rho   0.272 n=5: rho   0.225 (These are approximate and only a sanity check; they are consistent with rho being on the order of 1/n.)

Lemma 1039.1 (exact rho for f(z)=z^n). For f(z)=z^n, one has rho(f)=1.

Proof. We have |f(z)|<1 iff |z|^n<1 iff |z|<1. Thus L(f) is exactly the open unit disc. The largest disc contained in the open unit disc has radius 1 (take the unit disc itself, or any disc of radius 1-epsilon). Hence rho(f)=1.

Lemma 1039.2 (a completely elementary universal lower bound). Let f(z)=$\prod_{i=1}^n$ $z-z_i$ with |z_i|<=1 and n>=2. Then rho(f) >= 3^{-$n-1$}.

Proof. Pick any root z_1. Consider the open disc D := {z: |z-z_1| < r} with r=3^{-$n-1$}. For any z in D, we have |z-z_1|<r. For j!=1, use the triangle inequality: |z-z_j| <= |z-z_1| + |z_1-z_j| < r + 2. Since r<1 for n>=2, we have r+2 < 3, hence |z-z_j| < 3 for all j!=1. Therefore \[ |f(z)| = \prod_{i=1}^n |z-z_i| < r * 3^{n-1} = 1. \] So D is contained in L(f), and hence rho(f) >= r = 3^{-$n-1$}.

VERIFICATION * Lemma 1039.1 is immediate from the definition. * Lemma 1039.2 uses only triangle inequality and the factorization of f. * The numerical sanity check was produced by an explicit script; it is not used as proof.

FINAL UNRESOLVED (i) Strongest proved partial result here: an explicit universal lower bound rho(f) >= 3^{-$n-1$} (Lemma 1039.2), and exact value rho$z^n$=1 (Lemma 1039.1). Numerical sanity checks for z^n-1 suggest rho$z^n-1$ is about 1/n. (ii) First gap: prove or disprove a polynomial-in-n lower bound of the form rho(f) >= c/n for an absolute constant c>0 (or determine the correct order of the worst case). (iii) Top 3 next moves: 1. Improve the elementary bound by exploiting that not all factors |z-z_j| can be simultaneously close to their worst-case upper bound near a root; try to relate $\prod_{j!=i} |z_i-z_j|$ to n and then control f in a neighbourhood. 2. Search for explicit families with unusually small rho(f) (potential counterexamples to 1/n), e.g. roots distributed near the unit circle in a way that makes all components of L(f) thin. 3. Rigorous computational exploration for moderate n (say n<=10): optimize over root locations (subject to |z_i|<=1) to minimize rho(f), to guess extremal configurations. (iv) Minimal counterexample structure (if rho(f) is smaller than c/n): a sequence of polynomials whose sublevel set |f|<1 breaks into many thin components, each with small inradius, perhaps via roots near the unit circle spaced to create narrow corridors.