Abstract
В настоящей заметке мы получим необходимое и достаточное условие на тройку неотрицательных целых чисел a < b < c при выполнении которого многочлен Нюмена $$\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$$ имеет корень на единичном круге. Изпользуя это условие мы докажем, что для каждого $$d \geq 3$$ существует такое целое положительное число n > d, что многочлен Нюмена $$1+x+\dots+x^{d-2}+x^n$$ длины d не имеет корней на единичном круге.
Highlights
For a polynomial f ∈ C[x], let throughout m(f ) := min |f (z)||z|=1 be the minimal value of f on the unit circle
Let Nd be the set of Newman polynomials of length d, namely, Nd := {xk1 + · · · + xkd, where k1 < · · · < kd are nonnegative integers}
[8] Mercer proved that μ(d) > 0 for each d ≥ 3. This is equivalent to the fact that for each d ≥ 3 there is a Newman polynomial of length d which has no roots on the unit circle
Summary
|z|=1 be the minimal value of f on the unit circle. Motivated by some questions raised by Campbell, Ferguson, Forcade [2], and Smyth [11] Boyd in [1] studied the behavior of m(f ) for Newman polynomials f. In [8] Mercer proved that μ(d) > 0 for each d ≥ 3 This is equivalent to the fact that for each d ≥ 3 there is a Newman polynomial of length d which has no roots on the unit circle. In this note we describe which Newman polynomials of the form a c f (x) = ∑︁ xj + ∑︁ xj = 1 + · · · + xa + xb + · · · + xc, j=0 j=b where 0 ≤ a < b ≤ c, have roots on the unit circle and which do not have. Φ(2l + 1) log 2 the Newman polynomial 1 + x + · · · + xd−2 + xn ∈ Nd, with degree n = 2φ(2l+1)k, has no roots on the unit circle.
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