We construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials. Many new examples of such polynomials of minimal absolute trace (for given degree) are found. The computations are pushed to degrees that previously were unattainable, and one consequence is that the new examples form the majority of ...

Remark 0.1. We saw that the study of cohomology and equivariant cohomology of Grassmannians leads to interesting symmetric polynomials, namely, the Schur polynomials sλ(x) and sλ(x|t). These arise in contexts other than intersection theory and representation theory. For example, Griffiths asked which polynomials P in c1(E), . . . , cn(E) are positive whenever E is an ample vector bundle on an n...

Using a ‘reasonable’ measure in P(`1 ), the space of 2homogeneous polynomials on `1 , we show the existence of a set of positive (and independent of n) measure of polynomials which do not attain their norm in the vertices of the unit ball of `1 . Next we prove that, when n grows, the measure of the set of polynomials which attain their norm in a face of ‘high’ dimension of the unit ball tends t...

An extension of symmetric classical orthogonal polynomials is d-symmetric classical d-orthogonal polynomials, d being a positive number. These polynomials and their derivatives satisfy particular (d+1)-order recurrence relations. Many works deals with these families. Our purpose is to derive some more results for this class of polynomials. In fact, in terms of hypergeometric functions, we expre...

Approximation of positive polynomials by sums of squares.