[Paper Review] Geometric involutive bases for positive dimensional polynomial ideals and SDP methods

Greg Reid, Fei Wang+1 more|arXiv (Cornell University)|May 28, 2014

Polynomial and algebraic computation38 references3 citations

TL;DR

This paper advances symbolic-numeric geometric involutive bases for positive-dimensional polynomial ideals by integrating them with semidefinite programming (SDP) methods, enabling efficient computation of real radicals. It introduces degree-reduction techniques and inclusion tests to simplify intermediate systems, achieving accurate real radical bases through moment matrix rank stabilization and projection-based pruning.

ABSTRACT

Geometric involutive bases for polynomial systems of equations have their origin in the prolongation and projection methods of the geometers Cartan and Kuranishi for systems of PDE. They are useful for numerical ideal membership testing and the solution of polynomial systems. In this paper we further develop our symbolic-numeric methods for such bases. We give methods to explicitly extract and decrease the degree of intermediate systems and the output basis. Algorithms for the numerical computation of involutivity criteria for positive dimensional ideals are also discussed. We were also motivated by some remarkable recent work by Lasserre and collaborators who employed our prolongation projection involutive criteria as a part of their semi-definite based programming (SDP) method for identifying the real radical of zero dimensional polynomial ideals. Consequently in this paper we begin an exploration of the interaction between geometric involutive bases and these methods particularly in the positive dimensional case. Motivated by the extension of these methods to the positive dimensional case we explore the interplay between geometric involutive bases and the new SDP methods.

Motivation & Objective

• To extend geometric involutive basis methods to positive-dimensional polynomial ideals, addressing limitations in existing symbolic-numeric approaches.
• To develop degree-reduction techniques that simplify intermediate systems and output bases, improving numerical stability and efficiency.
• To explore the interplay between geometric involutive bases and recent SDP-based methods for computing the real radical of polynomial systems.
• To adapt Cartan’s involutivity criteria for positive-dimensional ideals in a numerical setting, enabling practical implementation.
• To provide a framework that combines prolongation-projection algorithms with moment matrix completion for real solution computation in positive-dimensional cases.

Proposed method

• Uses a geometric prolongation-projection algorithm to compute involutive bases, generalizing Cartan’s criteria to positive-dimensional ideals.
• Applies an inclusion test in Algorithm 4.1 to discard higher-degree redundant systems during iteration, reducing matrix size and computational cost.
• Employs moment matrix construction via the subroutine gen(ker M), where M is the moment matrix and ker M is its kernel, to compute generators.
• Uses rank stabilization of the moment matrix (rank(M) = dim ker gif(Q)) as a termination criterion, analogous to Lasserre’s zero-dimensional method.
• Extracts projected systems from the symbol space using dimension tables (e.g., Figure 5), enabling degree reduction and simplification.
• Applies normalization and substitution (e.g., z = 2) to simplify higher-degree generators, yielding minimal, interpretable bases.

Experimental results

Research questions

• RQ1 How can geometric involutive bases be effectively extended to positive-dimensional polynomial ideals, where traditional Gröbner basis methods face scalability issues?
• RQ2 What role do projection and degree-reduction techniques play in simplifying intermediate systems and improving numerical conditioning?
• RQ3 How can the interaction between geometric involutive bases and SDP-based moment matrix methods be formalized and leveraged for real radical computation?
• RQ4 To what extent can the rank stabilization condition of moment matrices serve as a termination criterion for involutive basis computation in positive-dimensional cases?
• RQ5 Can the inclusion test in Algorithm 4.1 reliably eliminate redundant, higher-degree systems without compromising completeness of the basis?

Key findings

• The algorithm successfully computes a geometric involutive basis for the real radical of a positive-dimensional system, with output {z−2, x(z−2), y(z−2), z(z−2), x²−y²+3}, verified by hand calculation.
• The moment matrix rank was found to be 7, and the kernel of gen(ker M) had dimension 13, indicating a well-conditioned intermediate system.
• After applying the inclusion test, only the ℓ=1 entry in the first column of Figure 5 was retained, reducing the system size and eliminating redundant components.
• The projected system for ℓ=2, degree 1, yielded a simplified generator z−2 after normalization and small-term elimination.
• Substitution of z=2 into degree-2 generators reduced them to x²−y²+3, confirming consistency and minimal form.
• Termination was achieved when d=5=r, and the output contained 5 generators, with the rank stabilization condition matching Lasserre’s criterion in the zero-dimensional case.