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Register a new userAlgebraic structure of the range of a trigonometric polynomial
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The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.
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Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums with products of trigonometric functions can get complicated and may not have a simple expressions in a number of cases. Some of these sums have interesting properties and can have amazingly simple value. However, only some of them are available in literature. We obtain a number of such sums using method of residues.
Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning commutation with rotation, continuation beyond the domain of univalence, and periodicity.
In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmunds inequality to the polar derivative of a polynomial.
In his 2006 paper, Jin proves that Kalantaris bounds on polynomial zeros, indexed by $m leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $mrightarrowinfty$. That is, given a degree $n$ polynomial $p(z)$ not vanishing at the origin and an error tolerance $epsilon > 0$, Jin proves that there exists an $m$ such that $frac{L_m}{rho_{min}} > 1-epsilon$, where $rho_{min} := min_{rho:p(rho) = 0} left|rhoright|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jins theorem. In fact, we prove the stronger theorem that this convergence is uniform in a sense, its rate depending only on $n$ and a few other parameters. We also give experimental results that suggest an optimal m of (asymptotically) $Oleft(frac{1}{epsilon^d}right)$ for some $d ll 2$. A proof of these results would show that Jins method runs in $Oleft(frac{n}{epsilon^d}right)$ time, making it efficient for isolating polynomial zeros of high degree.
In this short Note we show that the direct image sheaf R 1 $pi$ * (O X) associated to an analytic family of compact complex manifolds $pi$ : X $rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.
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