A Successive Resultant Projection for Cylindrical Algebraic Decomposition

TL;DR

This paper demonstrates the equivalence of two projection operators used in cylindrical algebraic decomposition, one by Brown and another by Yang, through an identity that clarifies their relationship.

Contribution

It establishes the theoretical equivalence between Brown's projection and Yang's successive resultant projection in CAD.

Findings

Proves the equivalence of two CAD projection operators.

Uses an identity to show the operators are interchangeable.

Provides a theoretical foundation for choosing projection methods.

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.