Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Polynomials

abdalaoui_2025_l_alpha_flatness_erdos_littlewood_s/: Claims that plus-minus-one polynomials are not L^alpha-flat for even alpha above two, hence that no ultraflat such sequence exists; contradicted by the accepted Theorem 1.1 of the OpenAI 2026 release.

abdalaoui_nadkarni_2016_class_littlewood_polynomials_that_are_not_l_flat/: Proves coefficient-frequency restrictions on L-alpha-flat Littlewood sequences and excludes even-degree palindromic sequences, giving restricted obstructions relevant to Problem 1150.

anon_2026_bernstein_density_proof_erdos_s_robust/: Claims a proof that for any bound C, arbitrary nodes in the interval admit labels no low-degree polynomial of small uniform norm can nearly interpolate.

balint_1960_proof_conjecture_erdos/: Proves Erdos's conjecture that the gaps between consecutive zeros of the derivative of a polynomial with equally spaced real zeros increase outward.

balister_2020_flat_littlewood_polynomials_exist/: Proves that for every degree at least 2 there is a plus-minus-one polynomial whose modulus on the unit circle stays within constant factors of the root of the degree.

basu_pollack_roy_2006_algorithms_real_algebraic_geometry/: Basu, Pollack, and Roy's foundational algorithms text for constructible sets, projection, and quantifier elimination.

bernstein_1931_limitation_values_polynomial_segment/: Records the degree and node convention, global asymptotic, and qualified weaker local bounds.

bombieri_2009_kahane_ultraflat_polynomials/: Constructs unimodular polynomials of degree n whose modulus on the unit circle is the square root of n up to an error of order n to the power 1/2 minus 1/9 plus epsilon, for every epsilon > 0, sharpening Kahane's ultraflat polynomials, and makes the construction effective.

borwein_erdelyi_2003_lower_bounds_merit_factors_trigonometric_polynomials_littlewood_classes/: Gives quantitative fourth-moment and maximum-modulus gaps for real trigonometric and conjugate-reciprocal Littlewood-type classes.

borwein_mossinghoff_2008_barker_sequences_flat_polynomials/: Barker autocorrelation identities and the conditional consequences of long Barker sequences for Littlewood flatness, Mahler measure, and L1 norms.

cook_2021_universality_minimum_modulus_random_trigonometric_polynomials/: Proves that the minimum modulus of a random trigonometric polynomial with centered sub-Gaussian coefficients, scaled by the degree, has the same exponential limit law as in the Gaussian case, in particular for plus or minus one coefficients.

danchenko_2007_lengths_lemniscates/: Proves that the lemniscate where a monic degree-n polynomial has modulus r raised to n has length at most two pi n r.

deboor_1978_conjectures_bernstein_erdos_optimal_nodes_polynomial_interpolation/: Proves Bernstein's and Erdős's conjectures on optimal Lagrange interpolation nodes: exactly one node system makes the Lebesgue function equioscillate, it alone minimizes the Lebesgue constant, and every other system has a local maximum below that constant.

do_2024_strong_law_large_numbers_real_roots/: Proves that for Kac random polynomials the number of real roots in the interval from minus one to one divided by log n tends almost surely to one over pi.

erdelyi_2018_asymptotic_distance_between_ultraflat_unimodular_polynomial_its_conjugate_reciprocal/: Derives sharp moment and derivative constraints separating any ultraflat unimodular polynomial from its conjugate reciprocal.

erdelyi_2020_do_flat_skew_reciprocal_littlewood_polynomials_exist/: Gives a simplified Rudin–Shapiro and approximation-theoretic construction of constant-factor flat Littlewood polynomials with near skew reciprocity.

erdelyi_2025_sequence_partial_sums_unimodular_power_series_is_not_ultraflat/: Proves that the successive partial sums of one fixed unimodular power series cannot form an ultraflat sequence, even for complex phases.

erdelyi_2026_erdos_problem_about_maximum_modulus_littlewood_polynomials_unit_circl/: Proves that every degree n Littlewood polynomial has maximum squared modulus at least n plus one plus one thirty-eighth times n to the one-third.

erdos_1941_divergence_properties_lagrange_interpolation_parabolas/: Proves that Lagrange interpolation at Chebyshev nodes diverges to infinity for some continuous function at the points cos(p pi/q) with p and q odd.

erdos_1947_remarks_polynomials/: Collects results on sums of a polynomial's critical values, on Lagrange interpolation sums, and on leading coefficients of integer polynomials.

erdos_1949_number_terms_square_polynomial/: Shows the least number of terms in the square of a real polynomial with k terms is at most a constant times k to a power below 1.

erdos_1952_sum/: Shows that for an irreducible integer polynomial f, positive on the positive integers, the sum of d(f(k)) over k up to x lies between two constant multiples of x log x.

erdos_1956_number_real_roots_random_algebraic_equation/: Shows that all but a proportion o((log log n)^{-1/2}) of the degree-n polynomials with plus-or-minus one coefficients have (2/pi) log n + o((log n)^{1/2} log log n) real roots.

erdos_1958_metric_properties_polynomials/: Studies the size, measure, components and convexity of the lemniscate set where a monic polynomial has modulus below one, and poses sixteen problems.

erdos_1961_extremal_problem_theory_interpolation/: Studies Hermite derivative-data extremality and the ordinary Lagrange global and local questions.

erdos_1961_problems_results_interpolation_ii/: Records the ordinary-Lagrange global improvement and the weaker historical local bound.

erdos_1962_inequality_maximum_trigonometric_polynomials/: Proves a quantitative gap above Parseval for a dense real trigonometric class and formulates the corresponding maximum-modulus conjecture.

erdos_1967_problems_results_convergence_divergence_properties_lagrange/: Discusses the divergence of Lagrange interpolation and poses extremal problems on the Lebesgue function of a node set.

erdos_1989_convergent_interpolatory_polynomials/: Characterizes the node systems for which every continuous function is interpolated by polynomials of degree at most (1+epsilon)n whose error is of the order of the best approximation.

erdos_szabados_1978_integral_lebesgue_function_interpolation/: Records the published qualitative local integral bound for arbitrary interpolation nodes.

erdos_turan_1940_on_interpolation_iii/: Erdős and Turán's 1940 interpolatory theory of orthogonal and strongly normal polynomials: bounds for the node polynomial, nth-root asymptotics, root spacing and root distribution with error terms, and Lemma IV on two adjacent fundamental polynomials.

eremenko_1994_extremal_problem_polynomials/: Proves that a monic degree n polynomial whose set of modulus at most one is connected has derivative at most 2 to the power 1/n minus 1 times n squared on that set, with equality only for a shifted Chebyshev polynomial.

eremenko_1999_length_lemniscates/: Proves the level set where a monic degree d polynomial has modulus one has length at most 9.173 times d.

fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles/: Shows that over nodes in [-1,1] the sum of squares of the Lagrange fundamental functions has least possible maximum 1 on [-1,1], attained for n >= 2 only at the roots of (1-x^2) times the derivative of the (n-1)st Legendre polynomial, and computes the sum at other classical nodes.

fryntov_2009_new_estimates_length_erdos_herzog/: Shows the lemniscate length of a monic degree-n polynomial is at most 2n+O(n^{7/8}), and is locally maximal at z^n-1.

gunther_schmidt_2015_merit_factors_polynomials_derived_from_difference_sets/: Computes the limiting merit factors of Littlewood polynomials built from Gordon–Mills–Welch, Sidelnikov and cyclotomic sets; every limit is finite.

gunther_schmidt_2016_l_q_norms_fekete_related_polynomials/: Computes every fixed even-moment limit for Fekete, shifted Fekete, and finite-field character families of Littlewood polynomials.

halasz_1973_result_salem_zygmund_concerning_random_polynomials/: Proves max |f_n| = sqrt(n log n) + O(sqrt(n/log n) log log n) almost surely for random plus-minus one coefficients.

hayman_lingham_2018_research_problems_function_theory/: A nine-chapter collection of open problems in complex function theory with progress updates on each.

hong_2026_strategy_proposal_covering_lemniscates_erdos_problem/: A self-described strategy note proposing a mean-centered partition approach to covering polynomial lemniscates by disks of total radius at most two.

jedwab_et_al_2012_littlewood_polynomials_small_l_4_norm/: Disproves the conjecture that (7/6)^(1/4) is the least asymptotic L4 to L2 ratio of Littlewood polynomials, and determines the least ratio attained by a generalized Fekete family.

katz_moore_2017_sequence_pairs_lowest_combined_autocorrelation_crosscorrelation/: Characterizes equality in the Pursley–Sarwate bound and computes correlation asymptotics for recursive Golay families relevant to E1150.

konyagin_1994_minimum_modulus_random_trigonometric_polynomials_coefficients/: Proves that a random plus-minus-one trigonometric polynomial with n terms has, with probability tending to one, minimum modulus at most n to the power minus one half plus epsilon, for every fixed epsilon > 0.

krishnapur_2025_area_polynomial_lemniscates/: Shows the minimal area of a degree-n monic polynomial lemniscate lies between c/log n and C/log log n, and bounds the inradius below.

letwin_2026_maxima_littlewood_polynomials_1_1/: Determines the almost sure lower envelope of the maximum of a random Littlewood polynomial on the interval from minus one to one.

lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros/: Proves that arch areas, maxima and slopes of a polynomial with equally spaced real zeros grow outward, and reproves Balint's inequality that the gaps exceed one.

odlyzko_2018_search_ultraflat_polynomials_plus_minus_one_coefficients/: Source record and research digest.

openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials/: Claims that the smallest maximum modulus of a length-N sign polynomial on the unit circle is (1+o(1)) sqrt N through all integer lengths, by quadratic-phase sampling of a bounded torus polynomial and defect-sensitive sign rounding; deduces unbounded binary merit factors; bears on Problems 1150, 228 and 230.

openai_2026_circulant_hadamard_conjecture/: Proves the circulant Hadamard conjecture, that real circulant Hadamard matrices have order 1 or 4, by a group-ring descent at two and alternating character products at the odd primes, formally verified here; deduces the Barker-length list 2, 3, 4, 5, 7, 11, 13, of which only the even-length exclusion is formally verified, which touches the Barker route in Problem 1150. The prose is unreviewed.

openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials/: A release manuscript claiming that for every eta > 0 and every large N some plus-minus-one polynomial with N consecutive coefficients has modulus between sqrt(N)/16 and (1+eta) sqrt(N) on the unit circle, by the companion's relaxed construction plus a correction wave on its gaps. Problems 1150, 228 and 230.

openai_2026_ultraflat_real_littlewood_polynomials/: Claims that for every epsilon in (0,1) and every large N there are N signs whose polynomial has modulus between (1-epsilon)sqrt N and (1+epsilon)sqrt N on the whole unit circle, by a capped near-unimodular wave construction on the circle and discrepancy rounding; bears on Problems 1150, 228 and 230.

rack_2015_optimal_cubic_lagrange_interpolation_extremal_node/: Describes explicitly all node systems of four points on [-1,1] that minimize the Lebesgue constant of cubic Lagrange interpolation.

schinzel_1987_number_terms_power_polynomial/: Proves the Renyi-Erdos conjecture that a bound on the number of terms of a power of a polynomial bounds the number of terms of the polynomial.

schinzel_2009_number_terms_power_polynomial/: Improves the lower bound for the number of terms of a power of a polynomial in terms of the number of terms of the polynomial, removing one logarithm from Schinzel's 1987 bound.

tao_2025_maximal_length_erdos_herzog_piranian_lemniscate/: Proves the Erdos-Herzog-Piranian conjecture for all sufficiently large degree, with the maximum attained only by z^n - 1 up to symmetry.

tao_2026_local_bernstein_theory_lower_bounds_lebesgue/: Localizes Bernstein theory to rectangles and deduces sharp lower bounds for Lebesgue constants of interpolation on subintervals.

vertesi_2013_paul_erdos_interpolation_problems_results_new/: Survey of Erdos's work on Lagrange interpolation, Lebesgue constants, mean convergence and degree-raising, with the later results the survey reports on his questions.

wang_2026_proposed_complete_solution_erdos_problem_1038/: Claims a computer-assisted determination of the infimum 1.8344... of the measure of the sublevel set where a monic real-rooted polynomial is below one.

yakir_2021_approximately_half_roots_random_littlewood_polynomial/: Proves that all but o(2^n) of the sign polynomials of degree n-1 have n/2 + o(n) roots inside the unit disk.

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