Positivity of continuous piecewise polynomials

Piecewise Certificates of Positivity for matrix polynomials

We show that any symmetric positive definite homogeneous matrix polynomial M ∈ R[x1, . . . , xn] admits a piecewise semi-certificate, i.e. a collection of identites M(x) = P j fi,j(x)Ui,j(x) T Ui,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semialgebraic subset Si, where R = ∪ri=1Si. This result generalizes to the setting of biforms. Some examples of c...

Positivity Of Trigonometri Polynomials

Deciding positivity of real polynomials

We describe an algorithm for deciding whether or not a real polynomial is positive semideenite. The question is reduced to deciding whether or not a certain zero-dimensional system of polynomial equations has a real root.

Positivity of Legendrian Thom polynomials

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity resul...

Multivariate piecewise polynomials

This article was supposed to be on `multivariate splines'. An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a `multivariate spline', resulted in the answer that a multivariate spline is a possibly smooth, piecewise polynomial function of several arguments. In particular, the potentially very useful thin-plate spline was thought to belo...