Abstract. We give a δ-constant criterion for equinormalizability of deformations of isolated (not necessarily reduced) curve singularities over smooth base spaces of dimension ≥ 1. For one-parametric families of isolated curve singularities, we show that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.

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We study several deformation functors associated to the normalization of a reduced curve singularity (X, 0) ⊂ (C n , 0). The main new results are explicit formulas, in terms of classical invariants of (X, 0), for the cotangent cohomology groups T i , i = 0, 1, 2, of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the Ae-codimension of a parametrized curve singularity, where Ae denotes the Mather-Wall group of left-right equivalence.

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Abstract. In this paper we give a matrix version of Handelman's Positivstellensatz [4], representing polynomial matrices which are positive definite on convex, compact polyhedra.Moreover, we propose also a procedure to find such a representation. As a corollary of Handelman's theorem, we give a special case of Schmüdgen's Positivstellensatz for polynomial matrices positive definite on convex, compact polyhedra.