  Real algebraic geometry provides certificates for the positivity of
polynomials on semi-algebraic sets by expressing them as a suitable combination
of sums of squares and the defining inequalitites. We show how Putinar's
theorem for strictly positive polynomials on compact sets can be applied in the
case of strictly positive piecewise polynomials on a simplicial complex. In the
1-dimensional case, we improve this result to cover all non-negative piecewise
polynomials and give explicit degree bounds.
