Polynomials and Exponential Sums: Extremal Problems, Density, Inequalities, and Zeros

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Abstract Polynomials pervade mathematics. Virtually every branch of mathematics,from algebraic number theory and algebraic geometry to applied analysis,Fourier analysis, and computer science, has its corpus of theory arising from the study of polynomials. This project intends to study polynomial inequalities and their applications in classical analysis, approximation theory, orthogonal polynomials, and number theory. The proposed research is about polynomials in a general sense, so it includes Chebyshev spaces, Markov spaces, Descartes systems, Muntz polynomials, rational function spaces, as well as polynomials with various constraints such as restricted zeros, integer coefficients, and nonnegative coefficients in various bases. The project continues several years of successful work on a variety of topics and describes various new directions. The polynomial is one of the most basic concepts of mathematics. Throughouthistory people have found problems concerning polynomials especially fascinating. Each of the ``three famous problems'' in the ``Heroic Age'' dealt with zeros of polynomials: squaring the circle, duplicating the cube, and trisecting an angle. Historically, questions relating to polynomials, for example, the solutionof polynomial equations, gave rise to some of the most important problems ofthe day. Besides the natural intellectual interest in them, polynomials, and inparticular extremal problems involving polynomials, arise not only inalmost every field of mathematics, but also in other sciences, especially in engineering. However, it is often the case that polynomials related to a practical problem belong to a restricted class. Depending on the nature of the problem, some additional pieces of information on the polynomial may be known. For filter design, polynomials with coefficients from {-1,0,1} are very useful, because, if they are used, then the filter can be implemented without the use of multiplications; their implementation requires only additions and subtractions. This reduces the cost of implementation. The problem is to find such polynomials which have a ``lowpass'' behavior on the unit circle of the complex plane. Roughly, that is, they approximate 1 in a neighborhood of 1 and approximate 0 in a neighborhood of -1. The project plans to study extremal properties of polynomials with coefficients from {-1,0,1}{0,1}, and {-1,1}. Unimodular polynomials (when the coefficients are complex numbers of modulus 1) are also important in engineering. In some other problems related to physics, in particular electrostatics, information about the distribution of the zeros of the polynomial is known. ``Gaped'' polynomials (polynomialswith a large number of zero coefficients) are used to analyze, for example,the time-optimal boundary controls for the one-dimensional heat equation. The proposal intends to explore further applications of polynomial inequalitiesas well as to offer a systematic study of naturally arising questions regarding polynomials subject to various constraints.