Zeta functions & hyperplane arrangements
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Zeta functions encode infinite sequences \((a(n))_n\):
\begin{aligned}
\zeta(s) &= 1 + \dfrac{1}{2^s} + \dfrac{1}{3^s} + \dfrac{1}{4^s} + \cdots \\
&= \sum_{n=1}^{\infty} 1 \cdot n^{-s}
\end{aligned}
Zeta functions shine light on arithmetic data:
Replace global zeta function with local zeta functions (primes).
What is it good for?
\zeta(s) = \prod_{p\text{ prime}} \dfrac{1}{1 - p^{-s}}
If \(\mathrm{gcd}(m, n) = 1\), then
a(m)a(n) = a(mn).
The Igusa zeta function
Let \(f\in \mathbb{Z}[x_1, \dots, x_d]\). For every positive integer \(n\):
R(n) = \#\left\{z \in \left(\mathbb{Z}/n\mathbb{Z}\right)^d ~\middle|~ f(z) \equiv 0\; (\text{mod }n) \right\}
- Can we write a formula for \(R(n)\)?
If \(n = p_1^{e_1} \cdots p_{\ell}^{e_{\ell}}\) is the prime factorization of \(n\), then
R(n) = R(p_1^{e_1}) \cdots R(p_{\ell}^{e_{\ell}})
So we focus on a prime \(p\).
- How does \(R(p_i^{e_i})\) vary as \(i\) varies?
- How does \(R(n)\) grow as \(n\) grows?
- Finite distinct formulae for the \(R(p^e)\) over all primes \(p\)?
We focus on polynomials equal to product of linear factors.
\begin{array}{r|cccc}
e & 1 & 2 & 3 & 4 & 5 & 10\\ \hline
R(p^e) & {\color{blue}9}p^2 & {\color{blue}22}p^4 & {\color{blue}7}p^7 & {\color{blue}3}p^{{\color{blue}10}} & {\color{blue}13}p^{12} & {\color{blue}9}p^{{\color{blue}26}}
\end{array}
Leading terms!
\begin{array}{r|cccc}
e & 1 & 2 & 3 & 4 & 5 & 10\\ \hline
R(p^e) & {\color{blue}6}p^2 & {\color{blue}13}p^4 & {\color{blue}4}p^7 & {\color{blue}13}p^{{\color{blue}9}} & p^{12} & {\color{blue}13}p^{{\color{blue}24}}
\end{array}
Theorem (Igusa 1974).
- The \(R(p^e)\) grow like a polynomial.
- Finite distinct formulae for the \(R(p^e)\).
- Finite linear recurrence relation among \((R(p^{e_i}))_i\).
Example.
f = x_1x_2x_3(x_1^2 - x_2^2)(x_1^2 - x_3^2)(x_2^2 - x_3^2)
- \(p >2\)
- \(p =2\)
Overview
Understand features of Igusa zeta function with combinatorial & topological tools
flag Hilbert–Poincaré series
Igusa zeta function
combinatorial "coarsening"
Define flag Hilbert–Poincaré series and give substitution.
Hyperplane arrangements
Hyperplane arrangement \(\mathcal{A}\) finite set of hyperplanes in \(K^d\).
Intersection poset: ordered by reverse inclusion
\mathcal{L}(\mathcal{A}) = \left\{\bigcap_{H\in S} H ~\middle|~ S\subseteq \mathcal{A} \right\}.
- Bottom element \(\hat{0}\) : the vector space \(K^d\).
- Top element \(\hat{1}\) : the common intersection (if exists).
If \(\hat{1}\in\mathcal{L}(\mathcal{A})\), then \(\mathcal{A}\) is central.
\pi_{\mathcal{A}}(Y) = \displaystyle\sum_{x\in\mathcal{L}(\mathcal{A})} |\mu(\hat{0},x)|\cdot Y^{\mathrm{codim}(x)} = \mathrm{Poin}(\mathbb{C}^d \setminus \bigcup_{H\in\mathcal{A}}H;\; Y)
Poincaré polynomial:
Coefficients are Betti numbers.
Example.
\displaystyle\mathsf{A}_n = \prod_{1\leqslant i < j \leqslant n+1} (x_i - x_j).
\pi_{\mathsf{A}_3}(Y) = 1 + 6Y + 11Y^2 + 6Y^3
\(\mathrm{codim}\)
0
1
2
3
\(\mathcal{L}(\mathsf{A}_3)\)
\(\hat{0}\)
\(\hat{1}\)
\(\mathsf{A}_3\)
\(K\!=\!\mathbb{C}\)
Write \(\Delta(\mathcal{L}(\mathcal{A})\setminus \{\hat{0}\})\) for flags of proper subspaces in \(\mathcal{L}(\mathcal{A})\).
For flag \(F = (F_1< \cdots <F_{\ell}) \in \Delta(\mathcal{L}(\mathcal{A})\setminus \{\hat{0}\})\), generalize:
\pi_F(Y) = \pi_{[\hat{0},F_1]}(Y) \pi_{[F_1, F_2]}(Y) \cdots \pi_{[F_{\ell}, \hat{1}]}(Y)
Example.
\(F = (\quad\;)\):
\begin{aligned}
\pi_F(Y) &= \pi_{[\phantom{m}, \phantom{m}]}(Y) \cdot\pi_{[\phantom{m}, \phantom{m}]}(Y) \\
&= (1+Y)\cdot (1 + 3Y + 2Y^2)
\end{aligned}
Self Reciprocity
The flag Hilbert–Poincaré series:
Theorem (M., Voll 2024). For \(\mathcal{A}\) defined over field of characteristic \(0\) and central, then
\mathsf{fHP}_{\mathcal{A}}\left(Y^{-1},(T_x^{-1})_{x\in\mathcal{L}(\mathcal{A})}\right) = (-Y)^{-\mathrm{rk}(\mathcal{A})} T_{\hat{1}} \cdot \mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}).
\mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}) = \displaystyle\sum_{F\in\Delta(\mathcal{L}(\mathcal{A})\setminus \{\hat{0}\})} \pi_F(Y) \prod_{x\in F} \dfrac{T_x}{1 - T_x}
Strong regularity via Poincaré duality \(\longrightarrow\) palindromic coeffs.
For \(x\in\mathcal{L}(\mathcal{A})\), set
\lambda(x) = \# \left\{ H\in\mathcal{A} ~\middle|~ x\subseteq H \right\}.
All potential poles come from combinatorial data:
\mathrm{Re}(s) \in \left\{ -\dfrac{\mathrm{codim}(x)}{\lambda(x)} ~\middle|~ x\in\mathcal{L}(\mathcal{A})\setminus\{\hat{0}\}\right\} \subseteq \mathbb{Q}\cap [-1,0)
Back to Igusa zeta functions
is the Igusa zeta function associated to \(\mathcal{A}\).
Theorem (M., Voll 2024). If \(\mathcal{A}\) defined over \(\mathbb{Q}\), then for all but finitely many primes \(p\),
\mathsf{fHP}_{\mathcal{A}}\left(-p^{-1}, (p^{-\mathrm{codim}(x) - \lambda(x) s})_{x\in\mathcal{L}(\mathcal{A})}\right)
Example.
\mathcal{\mathsf{B}_3} :~ x_1x_2x_3(x_1^2 - x_2^2)(x_1^2 - x_3^2)(x_2^2 - x_3^2)
\begin{align*}
\dfrac{1 - p^{3-s} \cdot \left.Z_{\mathsf{B}_3}(-3-s)\right|_{p\to p^{-1}}}{(1 - p^{3-s})} &= \dfrac{1 + 15p^{-s} - 23p^{1-s} + 7p^{2-s} + \cdots + p^{42-17s}}{(1 - p^{2-s})^2(1 - p^{7-3s})(1 - p^{10 - 4s})(1 - p^{24 - 9s})} \\[1.25em]
\end{align*}
Back to our original data:
= 1 + (9p^2 + \cdots)p^{-s} + (22p^4 + \cdots)p^{-2s} + \cdots
\(\mathcal{L}(\mathsf{B}_3)\)
Z_{\mathsf{B}_3}(s) = \dfrac{(1 - p^{-1})(1 - 8p^{-1} + 15p^{-2} + \cdots + 15p^{-6-9s} - 8p^{-7-9s} + p^{-8-9s})}{(1 - p^{-1-s})(1 - p^{-2-3s})(1 - p^{-2-4s})(1 - p^{-3-9s})}
\lambda(\phantom{m}) = 9
\lambda(\phantom{m}) = 1
\lambda(\phantom{m}) = 2
\lambda(\phantom{m}) = 3
\lambda(\phantom{m}) = 4
A different coarsening
Set each \(T_x=T\) — coarse flag Hilbert–Poincaré series:
Nice form:
\begin{align*}
\mathcal{N}_{\mathsf{B}_3}(Y, T) &= 1 + 9Y + 23Y^2 + 15Y^3 + 20T + 76Y T \\
&\quad + 76Y^2T + 20Y^3T + 15T^2 + 23Y T^2 + 9Y^2T^2 + Y^3T^2
\end{align*}
\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \dfrac{\mathcal{N}_{\mathcal{A}}(Y, T)}{(1 - T)^{\mathrm{rk(\mathcal{A})}}}
\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \displaystyle\sum_{F\in\Delta(\mathcal{L}(\mathcal{A})\setminus \{ \hat{0}\})} \pi_F(Y) \left(\dfrac{T}{1 - T}\right)^{\# F}
of \(\mathcal{L}(\mathcal{A})\)
codim of
maximal elt
Corollary. For every hyperplane arrangement \(\mathcal{A}\), the coefficients of \(\mathcal{N}_{\mathcal{A}}(Y, T)\) are non-negative.
\{\text{maximal flags of }\mathcal{L}(\mathcal{A})\} \times \{\text{subsets of }\{1,\dots, \mathrm{rk}(\mathcal{A})\}\}
Theorem (Dorpalen-Barry, M., Stump +2023). The coefficients of \(\mathcal{N}_{\mathcal{A}}(Y, T)\) count
together with two statistics on the pair.
\mathcal{N}_{\mathcal{A}}(Y, T) = \sum_{(M, S)} Y^{\# S} T^{\mathrm{stat}(M, S)}.
Numerator polynomial looks something like:
Summary
- Combinatorially defined substitution from flag Hilbert–Poincaré series to Igusa zeta function.
- Self-reciprocity of flag Hilbert–Poincaré series yield palindromicity of coefficients.
- Coefficients of numerator of coarse flag Hilbert–Poincaré series are non-negative.
flag Hilbert–Poincaré series
Igusa zeta function
combinatorial "coarsening"
Zeta functions & hyperplane arrangements
By Josh Maglione
Zeta functions & hyperplane arrangements
We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert–Poincaré series. We show how these rational functions are connected to local Igusa zeta functions and class counting zeta functions for certain graphical group schemes studied by Rossmann and Voll. We report on a general self-reciprocity result and a non-negativity result of the numerator polynomial under a coarsening, and we explore other connections within algebraic combinatorics. We report on joint works with Christopher Voll and with Galen Dorpalen-Barry and Christian Stump.
