共查询到20条相似文献,搜索用时 15 毫秒
1.
Peter Borwein Tamá s Erdé lyi 《Proceedings of the American Mathematical Society》2001,129(3):725-730
We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.
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Jason Bell 《Discrete Mathematics》2007,307(6):668-682
We show that each polynomial a(z)=1+a1z+?+adzd in N[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen-Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of a(z) belong to the real interval [-1,0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form. 相似文献
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《Applied Mathematics Letters》2007,20(6):671-675
This note is a self-contained proof of an interesting property of the double-step Newton method, applied to the computation of the largest or smallest zero of a real polynomial with all real zeros. It deals with what occurs when the iterates overshoot the zero. Our proof technique seems more transparent than the one usually found in the literature, and may make future extensions of the theorem possible. 相似文献
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Amir Dembo Bjorn Poonen Qi-Man Shao Ofer Zeitouni 《Journal of the American Mathematical Society》2002,15(4):857-892
Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.
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We consider some analytic behaviors (convexity, monotonicity and number of critical points) of the period function of period annuli of the potential system and focus on the case when g(x) is a polynomial whose roots are all real. The main contributions of this paper are twofold: (i) analytic behaviors are given for the period functions of period annuli surrounding one or more and simple or degenerate equilibria; (ii) as a nontrivial application of the general conclusions in (i), a purely analytical and shorter proof is provided for a result for the case degg=4 recently obtained by Chengzhi Li and Kening Lu with some help of computer algebra [Chengzhi Li, Kening Lu, The period function of hyperelliptic Hamiltonian of degree 5 with real critical points, Nonlinearity 21 (2008) 465-483]. 相似文献
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Consider the polynomial equation
where 0 <r
1 ⪯ {irt}2⪯... ⪯r
n All zeros of this equation lie on the imaginary axis. In this paper, we show that no two of the zeros can be equal and the
gaps between the zeros in the upper half-plane strictly increase as one proceeds upward. Also we give some examples of geometric
progressions of the zeros in the upper half-plane in casesn = 6, 8, 10. 相似文献
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Fengming Dong 《Journal of Graph Theory》2019,92(4):361-376
For a bridgeless graph , its flow polynomial is defined to be the function , which counts the number of nonwhere-zero -flows on an orientation of whenever is a positive integer and is an additive Abelian group of order . It was introduced by Tutte in 1950, and the locations of zeros of this polynomial have been studied by many researchers. This paper gives a survey on the results and problems on the study of real zeros of flow polynomials. 相似文献
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Let be the minimal length of a polynomial with coefficients divisible by . Byrnes noted that for each , and asked whether in fact . Boyd showed that for all , but . He further showed that , and that is one of the 5 numbers , or . Here we prove that . Similarly, let be the maximal power of dividing some polynomial of degree with coefficients. Boyd was able to find for . In this paper we determine for .
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Fuad Kittaneh 《Proceedings of the American Mathematical Society》2007,135(3):659-664
We apply some eigenvalue inequalities to the real parts of the Frobenius companion matrices of monic polynomials to establish new bounds and a majorization for the real parts of the zeros of these polynomials.
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Hanan Aljubran Maxim L. Yattselev 《Journal of Mathematical Analysis and Applications》2019,469(1):428-446
Let be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say , of random polynomials where are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that admits an asymptotic expansion of the form (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case admits an analogous expansion with the coefficients depending on the measure μ for (the leading order term and remain the same). 相似文献
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In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary
type polynomials. 相似文献