cyjax.donaldson.reduced_monomial_basis

cyjax.donaldson.reduced_monomial_basis(degree, poly_pows)

Generate power matrix for monomial basis modulo a polynomial.

Denote the full set of degree \(d_s\) (=``degree``) monomials on \(\mathbb{C}^{\mathrm{dim_proj}+1}\) as \(s_i(z)\).

On the set where the degree \(d_p\) homogeneous polynomial \(p(z) = 0\), the monomials \(s_i(z)\) become linearly dependent. This function implements a simple, although possibly not optimal, scheme to obtain a reduced basis of monomials on that set.

If the monomial degree is less than the degree of the polynomial, nothing has to be done.

Where the polynomial \(p(z)\) vanishes, any linear combination of basis monomials proportional to \(p\) vanishes. Let \(r_j(z)\) denote the degree \(d_s - d_p\) monomials. The linear dependencies introduced by \(p(z)=0\) are then

\[r_j(z) p(z) = 0 \,,\]

which is a sum of degree \(d_s\) monomials. This function produces a basis by removing one of these (linear dependent) monomials for each \(r_j\) from the full set of monomials \(s_j\).

Parameters:

poly_pows – 2d integer matrix, giving the monomials which make up the polynomial to be modded out from basis.

Returns:

2d integer matrix.