Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary first-order formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac

Abstract. Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting. We are interested in computing factorizations of nonnegative Laurent polynomials into sum of squares of polynomials. That is, let P (z) = n�

Two easy constructive methods are presented to compute tight wavelets frames for any given re nable function whose mask satis es the QMF or sub-QMF conditions in the multivariate setting. By applying one of our constructive methods, tight wavelet frames for multivariate box splines, e.g., bivariate box splines on a three or four direction mesh are constructed. Then a construction of tight wavelet frames which have the maximum vanishing moments is given. Another easy constructive method of compactly supported biframe pairs is shown. A connection to the 17th in the Hilbert famous list of 23 problems is pointed out. x1.