Estimating Sums of Convergent Series via Rational Polynomials

Sums of Squares of Polynomials with Rational Coefficients

We construct families of explicit polynomials f over Q that are sums of squares of polynomials over R, but not over Q. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions over Q. In the case of ternary quartics, we prove that our counterexamples to Sturmfels’ question are the only ones.

Rational Sums of Hermitian Squares of Free Noncommutative Polynomials

In this paper we consider polynomials in noncommuting variables that admit sum of hermitian squares and commutators decompositions. We recall algorithms for finding decompositions of this type that are based on semidefinite programming. The main part of the article investigates how to find such decomposition with rational coefficients if the original polynomial has rational coefficients. We sho...

Tail sums of convergent series of independent random variables

Bessel Polynomials and the Partial Sums of the Exponential Series

Let e k (x) denote the k-th partial sum of the Maclaurin series for the exponential function. Define the (n + 1) × (n + 1) Hankel determinant by setting Hn(x) = det[e i+j (x)] 0≤i,j≤n. We give a closed form evaluation of this determinant in terms of the Bessel polynomials using the method of recently introduced γ-operators.

Estimating Mixture Models via Mixtures of Polynomials

Mixture modeling is a general technique for making any simple model more expressive through weighted combination. This generality and simplicity in part explains the success of the Expectation Maximization (EM) algorithm, in which updates are easy to derive for a wide class of mixture models. However, the likelihood of a mixture model is non-convex, so EM has no known global convergence guarant...