Rational Functions

The main purpose of this package is to explicitly compute the flag Hilbert–Poincaré series and its specializations like Igusa'a local zeta function. We defer to Maglione–Voll for the details on the rational functions.

BraidArrangementIgusa

Input:

  • a positive integer.

Output:

  • the Igusa zeta function associated with the braid arrangement.

This is a specialized algorithm for the braid arrangement and is significantly faster than IgusaZetaFunction on the braid arrangement. This is based off of Lemma 5.14 of Maglione–Voll.

Example (Time comparison)

We compute the Igusa zeta function associated with $\mathsf{A}_6$ and record the time (on the same machine).

sage: %timeit _ = hi.BraidArrangementIgusa(6)
5.55 ms ± 7.58 μs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit _ = hi.IgusaZetaFunction(hi.CoxeterArrangement("A6"))
4.76 s ± 28 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

Thus for $\mathsf{A}_6$ BraidArrangementIgusa is about 1000 times faster than IgusaZetaFunction—of course the latter is also general purpose.

Example (Large example)

We compute a factorization of the denominator of the Igusa zeta function associated with $\mathsf{A}_{10}$.

sage: hi.BraidArrangementIgusa(10).denominator().factor()
(y*t - 1)^5 * (y*t^2 - 1)^2 * (y*t^3 - 1) * (y*t^4 - 1) * (y^2*t^3 - 1)^3 * (y*t^5 - 1) * (y^2*t^4 + y*t^2 + 1)^2 * (y^2*t^5 - 1)^2 * (y^2*t^5 + 1)^2 * (y^2*t^7 - 1) * (y^2*t^9 - 1) * (y^2*t^9 + 1) * (y^2*t^10 + y*t^5 + 1) * (y^2*t^11 - 1) * (y^4*t^12 + y^3*t^9 + y^2*t^6 + y*t^3 + 1) * (y^4*t^14 + y^2*t^7 + 1) * (y^4*t^18 + 1) * (y^6*t^24 + y^5*t^20 + y^4*t^16 + y^3*t^12 + y^2*t^8 + y*t^4 + 1) * (y^6*t^30 + y^3*t^15 + 1) * (y^8*t^44 + y^6*t^33 + y^4*t^22 + y^2*t^11 + 1)

Using a program to rewrite the finite geometric progressions in the denominator (like BRational), the denominator takes the following form:

(1 - y*t)^5*(1 - y^2*t^3)^3*(1 - y^3*t^6)^2*(1 - y^4*t^10)^2*(1 - y^5*t^15)*(1 - y^6*t^21)*(1 - y^7*t^28)*(1 - y^8*t^36)*(1 - y^9*t^45)*(1 - y^10*t^55)

CoarseFHPSeries

Input:

  • arrangement: a hyperplane arrangement. Default None.
  • matroid: a matroid. Default None.
  • poset: a poset. Default None.
  • R_label: a function from pairs of elements of a poset into the integers. Default None.
  • numerator: return only the numerator. Default False.
  • verbose: turn on print statements. Default False.
  • method: a string stating which method to use. Default recursion.

Output:

  • the coarse flag Hilbert–Poincaré series associated to the graded poset determined by the input.

If R_label is given, it is not verified to be an $R$-label. To use the $R$-label, set method='R-label'. This will use the formula developed in Dorpalen-Barry et al. from Corollary 2.22. In testing, this method seems to be about two times slower than the default recursion.

The coarse flag Hilbert–Poincaré series associated with a graded poset $P$ is defined to be:

[ \mathsf{cfHP}_{P} (Y, T) = \sum_{F\in\Delta(P \setminus \{\hat{0}\})} \pi_F(Y) \left(\dfrac{T}{1 - T}\right)^{|F|} = \dfrac{\mathcal{N}_{P}(Y, T)}{(1 - T)^{\mathrm{rk}(P)}}. ]

Example (Boolean arrangement)

We verify that the Boolean arrangement, $\mathcal{A}$, of rank $n$ satisfies the equation

[ \mathsf{cfHP}_{\mathcal{A}}(Y, T) = \dfrac{(1+ Y)^n E_n(T)}{(1 - T)^n}, ]

where $E_n(T)$ is the $n$th Eulerian polynomial. We set $n=6$ for this example.

sage: A = hi.CoxeterArrangement(["A1"]*6)
sage: A
Arrangement of 6 hyperplanes of dimension 12 and rank 6
sage: cfHP = hi.CoarseFHPSeries(A)
sage: cfHP.factor()
(T - 1)^-6 * (T + 1) * (Y + 1)^6 * (T^4 + 56*T^3 + 246*T^2 + 56*T + 1)

Example (Coxeter at ${\footnotesize Y=1}$)

We verify Theorem D of Maglione–Voll for the Coxeter arrangement of type $\mathsf{D}_5$. Thus, we will show that

[ \mathsf{cfHP}_{\mathsf{D}_5} (1, T) = 1920\cdot \dfrac{1 + 26T + 66T^2 + 26T^3 + T^4}{(1 - T)^5}, ]

sage: A = hi.CoxeterArrangement("D5")
sage: A
Arrangement of 20 hyperplanes of dimension 5 and rank 5
sage: cfHP = hi.CoarseFHPSeries(A)
sage: cfHP.factor()
(-1) * (T - 1)^-5 * (Y + 1) * (Y^4*T^4 + 397*Y^4*T^3 + 19*Y^3*T^4 + 3143*Y^4*T^2 + 3074*Y^3*T^3 + 131*Y^2*T^4 + 3239*Y^4*T + 15624*Y^3*T^2 + 8556*Y^2*T^3 + 389*Y*T^4 + 420*Y^4 + 9694*Y^3*T + 25826*Y^2*T^2 + 9694*Y*T^3 + 420*T^4 + 389*Y^3 + 8556*Y^2*T + 15624*Y*T^2 + 3239*T^3 + 131*Y^2 + 3074*Y*T + 3143*T^2 + 19*Y + 397*T + 1)

So we get exactly what we expect:

sage: (cfHP(Y=1)).factor()
(-1920) * (T - 1)^-5 * (T^4 + 26*T^3 + 66*T^2 + 26*T + 1)

Example (The path poset)

We define the path poset $P_4 = \{1,\dots, 4\}$ with the usual order $<$ of natural numbers. This is a graded poset with an $R$-label. We then compute the corresponding coarse flag Hilbert–Poincaré series. We note that there is no matroid whose lattice of flats is isomorphic to $P_n$. 

sage: P = Poset(DiGraph([(i, i+1) for i in range(1, 5)]))
sage: hi.CoarseFHPSeries(poset=P).factor()
(T - 1)^-4 * (Y + 1) * (Y*T + 1)^3

FlagHilbertPoincareSeries

Input:

  • arrangement: a hyperplane arrangement. Default None.
  • matroid: a matroid. Default None.
  • poset: a poset. Default None.
  • verbose: turn on print statements. Default False.

Output:

  • the flag Hilbert–Poincaré series associated to the graded poset determined by the input data.

The flag Hilbert–Poincaré series of a graded poset $P$ is defined to be:

[ \mathsf{fHP}_{P} (Y, \bm{T}) = \sum_{F\in\Delta(P \setminus \{\hat{0}\})} \pi_F(Y) \prod_{x\in F} \frac{T_x}{1 - T_x}. ]

Example (Lines through the origin again)

Because of the massive amount of variables in this function, we keep the number of hyperplanes small in this example. We compute the flag Hilbert–Poincaré series of the arrangement given by 5 lines passes through the origin in a $2$-dimensional vector space. 

sage: K = CyclotomicField(5)
sage: R.<X, Y> = PolynomialRing(K)
sage: f = X**5 - Y**5
sage: A = hi.PolynomialToArrangement(f)
sage: A
Arrangement of 5 hyperplanes of dimension 2 and rank 2
sage: hi.FlagHilbertPoincareSeries(A).factor()
(T6 - 1)^-1 * (T5 - 1)^-1 * (T4 - 1)^-1 * (T3 - 1)^-1 * (T2 - 1)^-1 * (T1 - 1)^-1 * (Y + 1) * (Y*T1*T2*T3*T4*T5 + 4*T1*T2*T3*T4*T5 - Y*T1*T2*T3 - Y*T1*T2*T4 - Y*T1*T3*T4 - Y*T2*T3*T4 - 3*T1*T2*T3*T4 - Y*T1*T2*T5 - Y*T1*T3*T5 - Y*T2*T3*T5 - 3*T1*T2*T3*T5 - Y*T1*T4*T5 - Y*T2*T4*T5 - 3*T1*T2*T4*T5 - Y*T3*T4*T5 - 3*T1*T3*T4*T5 - 3*T2*T3*T4*T5 + 2*Y*T1*T2 + 2*Y*T1*T3 + 2*Y*T2*T3 + 2*T1*T2*T3 + 2*Y*T1*T4 + 2*Y*T2*T4 + 2*T1*T2*T4 + 2*Y*T3*T4 + 2*T1*T3*T4 + 2*T2*T3*T4 + 2*Y*T1*T5 + 2*Y*T2*T5 + 2*T1*T2*T5 + 2*Y*T3*T5 + 2*T1*T3*T5 + 2*T2*T3*T5 + 2*Y*T4*T5 + 2*T1*T4*T5 + 2*T2*T4*T5 + 2*T3*T4*T5 - 3*Y*T1 - 3*Y*T2 - T1*T2 - 3*Y*T3 - T1*T3 - T2*T3 - 3*Y*T4 - T1*T4 - T2*T4 - T3*T4 - 3*Y*T5 - T1*T5 - T2*T5 - T3*T5 - T4*T5 + 4*Y + 1)

This is, indeed, equal to

[ \dfrac{1 + Y}{1 - T_6}\left(1 + 4Y + (1 + Y)\sum_{i=1}\dfrac{T_i}{1 - T_i}\right). ]

IgusaZetaFunction

Input:

  • arrangement: a hyperplane arrangement. Default None.
  • matroid: a matroid. Default None.
  • poset: a poset. Default None.
  • verbose: turn on print statements. Default False.

Output:

  • the Igusa zeta function associated to the graded poset determined by the input data.

For a compact discrete valuation ring $\mathfrak{o}$ and a polynomial $f\in \mathfrak{o}[X_1,\dots, X_d]$, Igusa's local zeta function associated with $f$ is

[ Z_f(s) = \int_{\mathfrak{o}^d} |f(\bm{X})|^s\, |\mathrm{d}\bm{X}|. ]

If $Q_\mathcal{A}$ is the defining polynomial of a hyperplane arrangement, then Igusa's local zeta function associated $\mathcal{A}$ is $Z_{Q_\mathcal{A}}(s)$. The output to IgusaZetaFunction is a bivariate function in $y = q^{-1}$ and $t = q^{-s}$, where $q$ is the cardinality of the residue field of $\mathfrak{o}$. This assumes good reduction; see Section 1.1 of Maglione–Voll.

For data not represented by such a hyperplane arrangement, we define the Igusa zeta function associated with a graded poset $P$ to be [ Z_P(s) = \mathsf{fHP}_P\left(-q^{-1}, \left(q^{-g_x(s)}\right)_{x\in P\setminus \{\hat{0}\}}\right) , ] where $g_x(s) = \mathrm{rank}(x) + \#\{a\in P \mid a\leqslant x \text{ and $a$ is an atom} \} \cdot s$. When $P$ is the intersection poset of a hyperplane arrangement $\mathcal{A}$, then $Z_P(s)=Z_{Q_{\mathcal{A}}}(s)$, which follows from Theorem B of Maglione–Voll.

Example (Uniform matroid)

We compute the Igusa zeta function associated with the uniform matroid $U_{3,5}$.

sage: M = matroids.Uniform(3, 5)
sage: M
U(3, 5): Matroid of rank 3 on 5 elements with circuit-closures
{3: {{0, 1, 2, 3, 4}}}
sage: hi.IgusaZetaFunction(matroid=M).factor()
(y - 1) * (y*t - 1)^-2 * (y^4*t^2 - 4*y^3*t^2 + 3*y^3*t + 6*y^2*t^2 - 12*y^2*t + 6*y^2 + 3*y*t - 4*y + 1) * (y^3*t^5 - 1)^-1

Example (coordinate hyperplanes)

We show that the Igusa zeta function of $f=x_1\cdots x_{5}$ factors in the expected way.

sage: A = hi.PolynomialToArrangement('*'.join(f'x{i}' for i in range(1, 6)))
sage: A.hyperplanes()
(Hyperplane 0*x1 + 0*x2 + 0*x3 + 0*x4 + x5 + 0,
 Hyperplane 0*x1 + 0*x2 + 0*x3 + x4 + 0*x5 + 0,
 Hyperplane 0*x1 + 0*x2 + x3 + 0*x4 + 0*x5 + 0,
 Hyperplane 0*x1 + x2 + 0*x3 + 0*x4 + 0*x5 + 0,
 Hyperplane x1 + 0*x2 + 0*x3 + 0*x4 + 0*x5 + 0)
sage: hi.IgusaZetaFunction(A).factor()
(y - 1)^5 * (y*t - 1)^-5

TopologicalZetaFuncion

Input:

  • arrangement: a hyperplane arrangement. Default None.
  • matroid: a matroid. Default None.
  • poset: a poset. Default None.
  • verbose: turn on print statements. Default False.

Output:

  • the topological zeta function associated with the graded poset determined by the input data.

For a graded poset $P$, the topological zeta function associated with $P$ is [ Z_{P}^{\mathrm{top}}(s) = \sum_{F\in \Delta(P\setminus\{\hat{0}\})} \pi_{P,F}^\circ(-1) \prod_{x\in F} \dfrac{1}{g_x(s)} , ]

where $g_x(s) = \mathrm{rank}(x) + \#\{a\in P \mid a\leqslant x \text{ and $a$ is an atom} \} \cdot s$ and [ \pi^{\circ}_{P, F}(Y) = \dfrac{\pi_F(Y)}{(1 + Y)^{\# F}}. ] When $P$ is the intersection poset associated with a hyperplane arrangement $\mathcal{A}$, then $Z_{P}^{\mathrm{top}}(s) = Z_{Q_{\mathcal{A}}}^{\mathrm{top}}(s)$, which follows from Corollaery 1.5 of Maglione–Voll.

Example (Shi arrangement)

We consider the Shi $\mathsf{A}_2$ arrangement and compute its topological zeta function. The Shi $\mathsf{A}_2$ arrangement is defined to be [ \mathcal{S} \mathsf{A}_2 = \left\{X_i - X_j - k ~\middle|~ 1\leqslant i < j\leqslant 3,\; k\in \{0,1\}\right\}. ]

sage: A = hi.ShiArrangement("A2")
sage: A
Arrangement of 6 hyperplanes of dimension 3 and rank 2

The topological zeta function is 

sage: Z = hi.TopologicalZetaFunction(A)
sage: Z.factor()
(s + 1)^-2 * (3*s + 2)^-1 * (12*s^3 + 2*s^2 - 5*s + 2)

which is [ Z_{\mathcal{S}\mathsf{A}_2}^{\mathrm{top}}(s) = \dfrac{2 - 5s + 2s^2 + 12s^3}{(s+1)^2(3s+2)} . ]