Computing the multiplicity structure from geometric involutive form

Abstract

Abstract. We present a method based on symbolic-numeric reduction to geometric involutive form to compute the primary component and the differential operators for an isolated singular solution of a polynomial ideal. The singular solution can be exact or approximate. If the singular solution is known with limited accuracy, then we propose a new method to refine it to high accuracy.

A note on geometric involutive bases for positive dimensional polynomial ideals and SDP methods

“…The dimension table is in Figure 2. Table of dim π ℓ D k (P 2 ) for system (41) The (blue) boxed 11 in the third column corresponds to π 2 D 2 (P 2 ). Now dim π 2 D 2 (P 2 ) = dim π 3 D 3 (P 2 ) so π 2 D 2 (P 2 ) satisfies one of the conditions for an involutive system.…”

Finding points on real solution components and applications to differential polynomial systems

“…Homotopy methods are used in [17] and [3] for real algebraic geometry. Lasserre et al [15] uses semi-definite programming and interestingly that approach is related to the prolongation-projection method used in geometrical completion of differential systems (also see Wu and Zhi [26]). …”

Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth one

“…Computing a closed basis of the local dual space is done essentially by matrix-kernel computations (Marinari et al, 1995;Mourrain, 1996;Dayton and Zeng, 2005;Wu and Zhi, 2008;Zeng, 2009), which are based on the stability property of Dx:…”