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Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let be the mth coefficient of the square f(x)² of a unimodular series , where all satisfy . We show that then and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that for each m ≥ m(ε).
Artūras Dubickas. "Heights of squares of Littlewood polynomials and infinite series." Annales Polonici Mathematici 105.2 (2012): 145-153. <http://eudml.org/doc/280396>.
@article{ArtūrasDubickas2012, abstract = {Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of a unimodular series $f(x) = ∑_\{i=0\}^\{∞\} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_\{m → ∞\} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε).}, author = {Artūras Dubickas}, journal = {Annales Polonici Mathematici}, keywords = {Littlewood polynomial; height; length; Hoeffding's inequality}, language = {eng}, number = {2}, pages = {145-153}, title = {Heights of squares of Littlewood polynomials and infinite series}, url = {http://eudml.org/doc/280396}, volume = {105}, year = {2012}, }
TY - JOUR AU - Artūras Dubickas TI - Heights of squares of Littlewood polynomials and infinite series JO - Annales Polonici Mathematici PY - 2012 VL - 105 IS - 2 SP - 145 EP - 153 AB - Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of a unimodular series $f(x) = ∑_{i=0}^{∞} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_{m → ∞} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε). LA - eng KW - Littlewood polynomial; height; length; Hoeffding's inequality UR - http://eudml.org/doc/280396 ER -