Open weak CAD and its applications

Abstract

The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea is to compute the intersection of projection factor sets produced by different projection orders. The resulting open weak CAD often has smaller number of sample points than open CADs.The algorithm can be used for computing sample points for all open connected components of f = 0 for a given polynomial f . It can also be used for many other applications, such as testing semi-definiteness of polynomials and copositive problems. In fact, we solved several difficult semidefiniteness problems efficiently by using the algorithm. Furthermore, applying the algorithm to copositive problems, we find an explicit expression of the polynomials producing open weak CADs under some conditions, which significantly improves the efficiency of solving copositive problems.

Multivariate discriminant and iterated resultant

“…It is worthwhile to note that, in practice, iterated resultants appear frequently in CAD (Cylindrical Algebraic Decomposition) [3], and our proof is based on the results about CAD we established in a series works [7,6,8]. Moreover, the polynomial Hp(f, [x 1 , .…”

“…[8] For generic form f (x n ), there exists a nonzero polynomial h ∈ Z[C α ], such that for any open connected set S ofR N \V R (Hp(f, [x n , . .…”

Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection

“…A key concept in CAD algorithm is the projection operator. Although many improved projection operators have been proposed [11,16,17,4,10,9,22], the CAD method is still of doubly exponential time complexity. The main reason is that in order to carry enough information, projection of variables causes the number of polynomials grows rapidly.…”