A Successive Resultant Projection for Cylindrical Algebraic Decomposition

Abstract

This note shows the equivalence of two projection operators which both can be used in cylindrical algebraic decomposition (CAD) . One is known as Brown's Projection (C. W. Brown (2001)); the other was proposed by Lu Yang in his earlier work (L.Yang and S.~H. Xia (2000)) that is sketched as follows: given a polynomial $f$ in $x_1,\,x_2,\,\cdots$, by $f_1$ denote the resultant of $f$ and its partial derivative with respect to $x_1$ (removing the multiple factors), by $f_2$ denote the resultant of $f_1$ and its partial derivative with respect to $x_2$, (removing the multiple factors), $\cdots$, repeat this procedure successively until the last resultant becomes a univariate polynomial. Making use of an identity, the equivalence of these two projection operators is evident.