Positive polynomials and numerical approximation
Bruno Després, Laboratoire J.-L. Lions, Université Paris 6

 The design of high order methods with strong control of the maximum principle appears to be still a core problem for the numerical approximation of many problems in scientific computing, including non linear equations.  In these directions, a natural question is  having good representations of polynomial with bounds. I will describe some recent ideas based on the Lukacs Theorem (and representation as some of squares) in order to design all such polynomials: it results in  a  new efficient Newton-Raphson algorithm which has been adapted  for the numerical solution of the advection equation. The case of polynomial with two bounds can be treated with an ad-hoc quaternion algebra. A selection of open problems will be described.