Section 5.1: A family of examples

Description

This family of systems is a modification of Example 5.1 in [Li & Zhi JCS 2012].
For any s ≥ 2, the system has s variables, s polynomials of degrees at most 3. The system has a root at the origin with multiplicity 2^s and breadth 2, i.e. its Jacobian matrix at the origin has co-rank 2.

Our method

We apply our parametric normal form method to compute an extension of the original system that completely deflates the root at the origin and contain new variables describing the multiplicity structure at the origin.
We demonstrate that this deflated system has at most s*(2^s-1) variables, which is linear in the multiplicity.

Note that the nil-index at the origin is 2^(s-1), so the size of the Macaulay multiplicity matrix is proportional to the number monomials in s variables of degree at most 2^(s-1).

• For any s ≥ 2, simply run
  
     > [Fnf,Vnf] = deflate51b(s);
  
  to compute the deflated systems using our new normal form (nf) approach.
  Note that for s ≥4 the Macaulay multiplicity matrix was too large to compute, so we only show here the normal form approach.

Computational Results