Polynomials

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E0114/: Asks whether the curve where a monic degree n complex polynomial has absolute value one is longest for the polynomial z to the n minus one.

E0115/: Asks whether a monic degree n polynomial whose set of modulus at most one is connected has derivative at most (1/2+o(1))n^2 there; Eremenko and Lempert proved it, and Erdős's exact bound n^2/2 fails for every n.

E0116/: Asks whether a monic polynomial with roots in the unit disc has modulus below one on a set of area at least an inverse power of n; proved by Pommerenke (1961), with a constant over log n by Krishnapur, Lundberg and Ramachandran.

E0119/: Asks whether the maximum modulus on the unit circle of the partial products of z minus a unimodular point is unbounded, exceeds a power of n, and has partial sums above n^(1+c); yes by Wagner, Beck, and Korsky with GPT 5.6-Pro.

E0228/: Asks whether every large n admits a degree n polynomial with plus or minus one coefficients whose modulus stays within fixed multiples of the root of n on the unit circle; yes for every n at least 2 by Balister et al. (2020).

E0230/: Asks whether a polynomial with unimodular coefficients has maximum modulus on the unit circle exceeding the square root of its degree by a constant factor.

E0485/: Asks whether the least possible number of nonzero terms in the square of a rational polynomial with exactly k nonzero terms tends to infinity as k grows.

E0509/: Asks whether the set where a monic nonconstant complex polynomial has modulus at most one can be covered by circles whose radii sum to at most two.

E0521/: Asks whether, for a random polynomial of degree n with independent sign coefficients, the count of real roots divided by log n tends almost surely to two over pi.

E0522/: Concerns the behavior of a random polynomial of degree n whose coefficients are chosen independently and uniformly from plus one and minus one.

E0523/: Concerns the behavior of a random polynomial of degree n whose coefficients are chosen independently and uniformly from plus one and minus one.

E0524/: The order of magnitude, for almost every real number in the unit interval, of the maximum on minus one to one of the polynomial built from its binary digits.

E0525/: Asks whether almost all degree n polynomials with coefficients plus or minus one dip below absolute value one on the unit circle, and how small that minimum is.

E0975/: Asks whether the sum of the number of divisors of the values of an irreducible integer polynomial up to X is asymptotic to a constant times X log X.

E1039/: The radius of the largest disc inside the region where a monic complex polynomial with all roots in the unit disc has absolute value below one.

E1041/: Examines a polynomial whose roots all lie strictly inside the unit disc and the shape of the region where its absolute value is below one.

E1114/: Concerns real polynomials of degree n whose roots are all real and form an arithmetic progression.

E1129/: Asks a question about how large the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one can be.

E1130/: Asks for the extreme behavior of sums of the fundamental Lagrange interpolation polynomials built from nodes in the interval from minus one to one.

E1131/: Asks for the least value, over n nodes in the interval from minus one to one, of the integral of the sum of squares of the Lagrange basis polynomials, and whether that least value is 2 minus (1+o(1))/n.

E1132/: Asks for the best possible bounds on the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one.

E1133/: Asks whether, for each positive constant, a small positive number exists making a stated property hold for all large degrees of polynomial interpolation.

E1150/: Asks whether every plus or minus one polynomial of degree n has maximum modulus on the unit circle above (1+c) times the square root of n for some fixed c>0; answered no by the OpenAI release's construction of 2026-09-23.

E1151/: Asks a question about the Lagrange interpolation polynomial of degree n minus one matching a function at n given nodes in the interval from minus one to one.

E1152/: Asks a question about fixed sets of n distinct interpolation nodes in the interval from minus one to one together with a tolerance tending to zero.

E1153/: Asks a question about the size of the fundamental Lagrange interpolation polynomials determined by n nodes in the interval from minus one to one.

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Extremal and approximation problems for polynomials, including coefficient and norm bounds, zero distribution, and random polynomials.

Site tags routed here: analysis, divisors, number theory, polynomials, probability.