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A new projection operator based on cylindrical algebraic decomposition (CAD)is proposed. The new operator computes the intersection of projection factorsets produced by different CAD projection orders. In other words, it computesthe gcd of projection polynomials in the same variables produced by differentCAD projection orders. We prove that the new operator still guaranteesobtaining at least one sample point from every connected component of thehighest dimension, and therefore, can be used for testing semi-definiteness ofpolynomials. Although the complexity of the new method is still doublyexponential, in many cases, the new operator does produce smaller projectionfactor sets and fewer open cells. Some examples of testing semi-definiteness ofpolynomials, which are difficult to be solved by existing tools, have beenworked out efficiently by our program based on the new method.

Constructing Fewer Open Cells by GCD Computation in CAD Projection