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Research Statement Christopher D Sinclair October 2006 Abstract My research lies at the intersection of number theory and random matrix theory I am interested in the connection between heights of polynomials and ensembles of asymmetric random matrices While there are known links between number theory and random matrix theory e g statistical similarities between the statistics of the zeros of the Riemann zeta functions and the eigenvalues of certain ensembles of random matrices my research demonstrates a new connection between the two fields 1 Number Theory and Volumes of Polynomials Given a polynomial in C x its Mahler measure is the absolute value of the leading coefficient times the product of the absolute values of its roots outside the unit circle Mahler measure is a height measure of complexity of polynomials which for integer polynomials can be interpreted as cyclotomicness the Mahler measure of an irreducible integer polynomial is 1 if and only if the polynomial is cyclotomic It is still unknown if 1 is a limit point of Mahler measures of integer polynomials this is Lehmer s Problem 11 There are other open questions regarding the range of Mahler measure restricted to integer polynomials 1 1 1 Counting Reciprocal Polynomials with Bounded Mahler Measure A reciprocal polynomial is one whose coefficient vector is invariant under reversal of order In 1971 Smyth proved that if the Mahler measure of an irreducible integer polynomial is less than 1 3 then the polynomial is necessarily reciprocal x 1 is the sole exception It is known that for every positive integer N and T 0 there are a finite number of reciprocal integer polynomials of degree at most N and Mahler measure at most T We will call this number MN T In my thesis I use geometry of numbers techniques to give asymptotic estimates for MN T by computing the volume of coefficient vectors of reciprocal real polynomials of degree at most N and Mahler measure at most T 18 We will call this number VN T It turns out



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