Module localpoly

cvolume.localpoly.Nlocal(g, n, stratum, labeled=False, mode='derivative')

Return a local polynomial \(N_{g,n}^{\kappa}(b_1, ..., b_n)\), where \(\kappa\) is stratum.

INPUT:

• g – int, genus
• n – int, number of boundaries
• stratum – list or tuple, orders of zeros of the stratum
• labeled – boolean (default False), whether to consider labeled zeros or not
• mode – ‘derivative’ (default) or ‘recursive’, ‘derivative’ uses Arbarello-Cornalba formulae and ‘recursive’ uses recursion on local polynomials dervied from the same AC formulae

OUTPUT:

• a symmetric Sage Multivariate Polynomial in variables b1, b2, … .

EXAMPLES:

Here we compute local polynomial of genus 2, two bondary components and stratum \(\mathcal{Q}(5, 1, 1, 1)\):

sage: from cvolume import Nlocal
sage: S = PolynomialRing(QQ,['b%d' % i for i in range(1,10)])
sage: b1,b2,b3,b4,b5 = S.gens()[:5]
sage: Nlocal(2, 2, (5, 1, 1, 1))
1/192*b1^6 + 95/3072*b1^4*b2^2 + 95/3072*b1^2*b2^4 + 1/192*b2^6

And here is the same local polynomial, but with labeled zeros:

sage: Nlocal(2, 2, (5, 1, 1, 1), labeled = True)
1/32*b1^6 + 95/512*b1^4*b2^2 + 95/512*b1^2*b2^4 + 1/32*b2^6

Another example for a different stratum:

sage: Nlocal(2, 2, (7, 1))
35/192*b1^4 + 35/64*b1^2*b2^2 + 35/192*b2^4

Here we show that recursive and derivative methods give consistent answers:

sage: assert Nlocal(0, 5, [3, 1, 1, 1], mode = 'recursive') == Nlocal(0, 5, [3, 1, 1, 1])
sage: assert Nlocal(1, 3, [3, 1, 1, 1], mode = 'recursive') == Nlocal(1, 3, [3, 1, 1, 1])
sage: assert Nlocal(0, 4, [3, 1, 1, -1], mode = 'recursive') == Nlocal(0, 4, [3, 1, 1, -1])
sage: assert Nlocal(0, 3, [3, 1, -1, -1], mode = 'recursive') == Nlocal(0, 3, [3, 1, -1, -1])

cvolume.localpoly.shift(variables, k)

Return a copy of the list of variables shifted k units to the left.

cvolume.localpoly.stratum_to_F(g, n, stratum)

Return partition function Fs corresponding to the stratum. Function is truncated at a minimal admissible weight.

cvolume.localpoly.vanish(variables, indices)

Return a copy of the list of variables, in which elements at given indices are replaces by 0.