Open Weak CAD and its Applications

TL;DR

This paper introduces open weak CAD, an efficient variation of CAD, with an algorithm that computes projection polynomials by intersecting different projection orders, enabling applications in polynomial semi-definiteness and copositive problems.

Contribution

The paper proposes a novel algorithm for computing open weak CADs using intersection of projection factor sets, improving efficiency in applications like semi-definiteness testing and copositive problems.

Findings

Open weak CADs often have fewer sample points than open CADs.

The algorithm efficiently solves semi-definiteness problems.

Application to copositive problems yields explicit polynomial expressions.

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\neq0$ 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 semi-definiteness 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.