### Abstract

In this paper, we do the same for Lazard's method. This takes in a lex-least invariant CAD of $\RR^{n-1}$ as input and outputs a sign invariant CAD of $\RR^n$: consequently, it cannot be used recursively, but only for $x_n$, the first variable to be projected. In the further steps of the projection phase, we use Lazard's original projection operator. Nonetheless, reducing the output in the first step has a domino effect throughout the remaining steps, which significantly reduces the complexity. The long-term goal is to find a general projection operator that takes advantage of the equality constraint and can be used recursively, and this operator gives an important first step in that direction.

Original language | English |
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Title of host publication | Proceedings SC2 2019 |

Number of pages | 8 |

Publication status | Published - Sep 2019 |

### Cite this

*Proceedings SC2 2019*

**On Benefits of Equality Constraints in Lex-Least Invariant CAD (Extended Abstract.** / Nair, Akshar Sajive; Davenport, James; Sankaran, Gregory.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings SC2 2019.*

}

TY - GEN

T1 - On Benefits of Equality Constraints in Lex-Least Invariant CAD (Extended Abstract

AU - Nair, Akshar Sajive

AU - Davenport, James

AU - Sankaran, Gregory

PY - 2019/9

Y1 - 2019/9

N2 - There are two relevant methods for CAD: McCallum [1984] which used order invariant CAD's and Lazard [Lazard1994, McCallumetal2019] which used lex-least invariant CADs, and doesn't have the nullification problem of McCallum [1984]. McCallum [1999] was the first to prove a CAD operator based on McCallum [1984], that took advantage of an equational constraint.In this paper, we do the same for Lazard's method. This takes in a lex-least invariant CAD of $\RR^{n-1}$ as input and outputs a sign invariant CAD of $\RR^n$: consequently, it cannot be used recursively, but only for $x_n$, the first variable to be projected. In the further steps of the projection phase, we use Lazard's original projection operator. Nonetheless, reducing the output in the first step has a domino effect throughout the remaining steps, which significantly reduces the complexity. The long-term goal is to find a general projection operator that takes advantage of the equality constraint and can be used recursively, and this operator gives an important first step in that direction.

AB - There are two relevant methods for CAD: McCallum [1984] which used order invariant CAD's and Lazard [Lazard1994, McCallumetal2019] which used lex-least invariant CADs, and doesn't have the nullification problem of McCallum [1984]. McCallum [1999] was the first to prove a CAD operator based on McCallum [1984], that took advantage of an equational constraint.In this paper, we do the same for Lazard's method. This takes in a lex-least invariant CAD of $\RR^{n-1}$ as input and outputs a sign invariant CAD of $\RR^n$: consequently, it cannot be used recursively, but only for $x_n$, the first variable to be projected. In the further steps of the projection phase, we use Lazard's original projection operator. Nonetheless, reducing the output in the first step has a domino effect throughout the remaining steps, which significantly reduces the complexity. The long-term goal is to find a general projection operator that takes advantage of the equality constraint and can be used recursively, and this operator gives an important first step in that direction.

M3 - Conference contribution

BT - Proceedings SC2 2019

ER -