Degree bounds and lifting techniques for triangular sets

We study the representation of the solutions of a polynomial system by triangular sets, and concentrate on the positivedimensional case. We reduce to dimension zero by placing the free variables in the base-field, so the solutions can be represented by triangular sets with coefficients in a rational function field. First, we give bounds on the degree of these coefficients; then we show how to apply lifting techniques in this context, and point out the role played by the evaluation properties of the input system. Our algorithms are implemented in Magma; we present two applications. 1. MAIN RESULTS We first define triangular sets over a ring A. A triangular set is a family of polynomials T = (T1,..., Tn) in A[X1,..., Xn] such that, for k ≤ n: Tk depends only on (X1,..., Xk), T