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Code for computations involving Witten's r-spin class.
Authors: Felix Janda (main author), Aaron Pixton (code improvements), Johannes Schmitt (integration into admcycles)
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    Returns the polynomial limit of Witten's r-spin class in genus g with input Avector,
    as discussed in the appendix of [Pandharipande-Pixton-Zvonkine '16].

    More precisely, Avector is expected to be a vector of linear polynomials in QQ[r],
    with leading coefficients being rational numbers in the interval [0,1] and constant
    coefficients being integers. Elements of Avector should sum to C * r + 2g-2 for some
    nonnegative integer C. Then Wittenrspin returns the tautological class obtained as
    the limit of

    r^(g-1+C) W_{g,n}^r(Avector)

    for r >> 0 sufficiently large and divisible, evaluated at r=0.

    INPUT:

    - ``g`` -- integer; underlying genus

    - ``Avector``   -- tuple ; a tuple of either integers, elements of a polynomial
      ring QQ[r] in some variable r or tuples (C_i, D_i) which are interpreted as
      polynomials C_i * r + D_i. For the entries with C_i = 1 we assume D_i <=-2.

    - ``r_coeff``  -- integer or None (default: `None`); if a particular integer
      r_coeff = r is specified, the function will return the (unscaled) class W_{g,n}^r(Avector),
      not taking a limit for large r.

    - ``d`` -- integer; desired degree in tautological ring; will be set to g-1+C by
      default

    - ``rpoly``  -- bool (default: `False`); if True, return the limit of
      r^(g-1+C) W_{g,n}^r(Avector) without evaluating at r=0, as a tautclass with
      coefficients being polynomials in r

    EXAMPLES:

    We start by verifying the conjecture from the appendix of [Pandharipande-Pixton-Zvonkine '16]
    for g = 2 and mu = (2)::

      sage: from admcycles import Wittenrspin, Strataclass
      sage: H1 = Wittenrspin(2, (2,))
      sage: H2 = Strataclass(2, 1, (2,))
      sage: (H1-H2).is_zero()
      True

    We can also verify a new conjecture for classes of strata of meromorphic differentials for
    g = 1 and mu = (3,-1,-2). The argument (1/2,-1) stands for an insertion 1/2 * r -1::

      sage: H1 = Wittenrspin(1, (3, (1/2,-1), (1/2,-2)))
      sage: H2 = Strataclass(1, 1, (3,-1,-2))
      sage: (H1+H2).is_zero()
      True

    As a variant of this, we also verify that insertions r-b stand for poles of order b with
    vanishing residues::

      sage: R.<r> = PolynomialRing(QQ,1)
      sage: H1 = Wittenrspin(1, (5,r-2, r-3))
      sage: H2 = Strataclass(1, 1, (5,-2,-3), res_cond=(2,))
      sage: (H1+H2).is_zero()
      True

    We can also compute the (scaled) Witten's class without substituting r=0::

      sage: Wittenrspin(1,(2,r-2),rpoly=True)
      Graph :      [1] [[1, 2]] []
      Polynomial : (1/12*r^2 - 5/24*r + 1/12)*(kappa_1)_0 + (-1/12*r^2 + 29/24*r - 37/12)*psi_1 + (-1/12*r^2 + 17/24*r - 13/12)*psi_2
      <BLANKLINE>
      Graph :      [0] [[4, 5, 1, 2]] [(4, 5)]
      Polynomial : -1/48*r + 1/24
      <BLANKLINE>
      Graph :      [0, 1] [[1, 2, 4], [5]] [(4, 5)]
      Polynomial : 1/12*r^2 - 17/24*r + 13/12

    Instead of calculating the asymptotic, polynomial behaviour of Witten's class,
    we can also input a concrete value for r, using the option r_coeff. Below we
    verify the CohFT property of Witten's class, first for a separating boundary
    divisor::

      sage: from admcycles import StableGraph
      sage: A = Wittenrspin(2,(1,1),r_coeff=4)
      sage: gr = StableGraph([1,1],[[1,2,3],[4]],[(3,4)])
      sage: pb = gr.boundary_pullback(A)
      sage: vector(pb.totensorTautbasis(1)[1])
      (0, 3/4, 0, 3/4, 3/4)
      sage: B = Wittenrspin(1,(1,1,2),r_coeff=4)
      sage: B.basis_vector()
      (0, 1/4, 0, 1/4, 1/4)
      sage: C = Wittenrspin(1,(0,),r_coeff=4)
      sage: C.basis_vector()
      (3)

    Then for a nonseparating boundary divisor::

      sage: B = Wittenrspin(2,(2,),r_coeff=4)
      sage: gr = StableGraph([1],[[1,2,3]],[(2,3)])
      sage: pb = gr.boundary_pullback(B)
      sage: pb.totensorTautbasis(1)
      (0, 1/2, 1/4, 1/4, -1/4)
      sage: A1 = Wittenrspin(1,(2,1,1),r_coeff=4)
      sage: A2 = Wittenrspin(1,(2,0,2),r_coeff=4)
      sage: A3 = Wittenrspin(1,(2,2,0),r_coeff=4)
      sage: L=[t.basis_vector() for t in [A1,A2,A3]]; L
      [(0, 1/2, 0, 0, 1/4), (0, 0, 0, 1/4, -1/4), (0, 0, 1/4, 0, -1/4)]
      sage: sum(L)
      (0, 1/2, 1/4, 1/4, -1/4)


    We can also check manually, that interpolating the above results for
    large r precisely gives the output of the option rpoly=True::

      sage: from admcycles.double_ramification_cycle import interpolate
      sage: H1 = Wittenrspin(1, (3, (1/2,-1), (1/2,-2)), rpoly=True)
      sage: H1.basis_vector()
      (0, -1/4*r^2 + 5/2*r - 3, 1/4*r^2 - r, 1/4*r^2 - r - 3, -1/4*r^2 + 1/2*r + 5)
      sage: res=[]
      sage: pts = list(range(6,11,2))
      sage: for r in pts:
      ....:  res.append(r**(1-1+1)*Wittenrspin(1, (3, 1/2*r-1, 1/2*r-2),r_coeff=r).basis_vector(1))
      sage: v = vector([interpolate(pts, [a[i] for a in res],'r') for i in range(5)])
      sage: v-H1.basis_vector()
      (0, 0, 0, 0, 0)
    NcSsg|]}t|t�r|�qSr")�
isinstancer�r8rr"r"r#r:/r;zWittenrspin.<locals>.<listcomp>rr%rGrzEEntries of Avector must be Integers, Polynomials or tuples (C_i, D_i)c3s&|]}�|dko�|dkVqdS)r%r<Nr"r7)�Cvector�Dvectorr"r#�	<genexpr>Rr;zWittenrspin.<locals>.<genexpr>css|]}|��VqdS)N)�denom)r8�cr"r"r#rxUr;r'cSsg|]}t|��qSr")rrur"r"r#r:Zr;)�sm�rt�ctrsr<)rbrdrJr&c3s|]}|j�d�VqdS)r&N)r )r8�b)rGr"r#rxsr;)�moduli) rC�parentr(rrrrrt�numbers�IntegralrDrrr�
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rr�from_vector)rF�Avectorr`r!�rpolyrrHZpolyentriesr0rZAvectorXrra�Cry�moddictZmoduZrvectr")rvrwrGr#�Wittenrspin�sv{

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���r�cCs�td�\}}}|dkr"dd|Std�dtd|||d|t|d�j|dd�|d|�td�d|||dt|d|d|||d|||d|t|d�j|dd�|d|d���S)a�
    Returns the expression P_m(X,a) as defined in [Pandharipande-Pixton-Zvonkine '16, Section 4.5].

    TESTS::

      sage: from admcycles.witten import Pmpolynomial
      sage: Pmpolynomial(0)
      1
      sage: Pmpolynomial(1)
      -1/12*X^2 + 1/2*(X - 1)*a - 1/2*a^2 + 5/24*X - 1/12
      sage: Pmpolynomial(2)
      1/288*X^4 - 1/12*(5*X - 1)*a^3 + 1/8*a^4 + 7/288*X^3 + 1/48*(20*X^2 - 5*X - 4)*a^2 - 29/384*X^2 - 1/48*(6*X^3 + X^2 - 9*X + 2)*a + 7/288*X + 1/288
    zb,X,arr%r'r�)rrrrr r_)r-r~rrr"r"r#rzs^X�r)NNFrs)5�__doc__�collections.abcrr]r�Zadmcycles.admcyclesrZadmcycles.DRrrrrrr	r
rrr
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