Real limit points of quasi-componenets of regular chains

Abstract

The work reported here is motivated by problems arising in solving polynomial systems over the real numbers.

Irredundant Triangular Decomposition

“…(2), p n−m, j ∈ I (C) for every 0 j n. Since W n−m = Z (p n−m ), we have C ⊂ W n−m . Moreover, C is an irreducible component of W n−m since dim C = n − m due to (1). is proves the lemma.…”

“…Due to (3), there exists q j ∈ C[x S ] such that q j p n−m, j ∈ I(∆ i ) ⊂ I (C) for every 0 j n. Since I (C) is prime and (1). is proves the lemma.…”

“…(2) e second difficulty is to factor out components that are embedded in higher dimensional irreducible components, similarly as it is stated in the Ri problem, mentioned above. is problem has only been solved for triangular sets in one and two dimensions [6,1]. e second result of this paper is that we show how to use the results in [13] and turn their irredundant equidimensional decomposition into an irredundant triangular decomposition.…”

“…. en (1) e system f 1 = f 2 = f 3 = 0 defines an irreducible twodimensional variety, so d 0 = d 1 = 2. e output of [13] is a set of at most 5 polynomials defining this irreducible variety, so we can assume that…”