[Paper Review] Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics

Botong Wang|arXiv (Cornell University)|Sep 12, 2015

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper computes the maximum likelihood degree (MLD) of Fermat hypersurfaces $F_{n,d} = \{x_0^d + \cdots + x_n^d = 0\} \subset \mathbb{P}^n$ using topological methods, specifically via the signed Euler characteristic of $F_{n,d} \setminus \mathcal{H}$, where $\mathcal{H}$ is the union of coordinate and sum hyperplanes. The key result expresses MLD in terms of combinatorial constants $\beta_{\mu,\nu}$, counting complex solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $z_1 + \cdots + z_\mu + 1 = 0$, with closed forms derived when $d-1$ is a prime power.

ABSTRACT

Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We give a formula of the Maximum likelihood degree in terms of the constants $β_{μ, ν}$, which is defined to be the number of complex solutions to the system of equations $z_1^ν=z_2^ν=\cdots=z_μ^ν=1$ and $z_1+\cdots +z_μ+1=0$.

Motivation & Objective

• To systematically compute the maximum likelihood degree (MLD) of Fermat hypersurfaces $F_{n,d}$, overcoming limitations of prior case-by-case approaches.
• To resolve the issue of non-transverse intersections in the general MLD formula by computing correction terms via Milnor number theory.
• To establish a topological formula for MLD in terms of Euler characteristics of $F_{n,d} \setminus \mathcal{H}$, leveraging Huh's result that MLD equals the signed Euler characteristic for smooth varieties.
• To provide explicit formulas for $\beta_{\mu,\nu}$, the number of complex solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $z_1 + \cdots + z_\mu + 1 = 0$, particularly when $d-1$ is a prime power.
• To connect the MLD computation to number-theoretic and combinatorial structures via the constants $\alpha_{\mu,\nu}$, defined as solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $z_1 + \cdots + z_\mu = 0$.

Proposed method

• Uses Huh's theorem that for smooth projective varieties, the MLD equals the signed Euler characteristic $(-1)^n \chi(F_{n,d} \setminus \mathcal{H})$.
• Applies the inclusion-exclusion principle to compute $\chi(F_{n,d} \setminus \mathcal{H})$ as a sum over intersections with hyperplanes in $\mathcal{H}$, indexed by subsets of the index set $\Lambda = \{0,\dots,n, +\}$.
• Analyzes singularities of the intersections $F_{n,d} \cap H_{\Lambda'}$ using Milnor theory, showing that each isolated singularity has Milnor number equal to the number of solutions counted by $\beta_{\mu,\nu}$.
• Relies on Milnor's result that the Euler obstruction at an isolated hypersurface singularity is $(-1)^{\dim} (\mu + 1)$, where $\mu$ is the Milnor number.
• Translates the combinatorial constants $\beta_{\mu,\nu}$ into symmetric group actions and multinomial coefficients when $\nu = p^r$ is a prime power.
• Uses the relation $\beta_{\mu,\nu} = \frac{1}{\nu} \alpha_{\mu+1,\nu}$ to reframe the problem in terms of symmetric solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $\sum z_i = 0$, enabling group-based counting.

Experimental results

Research questions

• RQ1 What is the maximum likelihood degree of the Fermat hypersurface $F_{n,d} \subset \mathbb{P}^n$ defined by $x_0^d + \cdots + x_n^d = 0$?
• RQ2 How can the MLD be computed in a systematic way when the general formula from [CHKS] fails due to non-transverse intersections?
• RQ3 What is the precise topological contribution of singularities arising from intersections of $F_{n,d}$ with coordinate and sum hyperplanes?
• RQ4 Under what conditions is $\beta_{\mu,\nu} \neq 0$, and can a closed-form expression be derived for $\beta_{\mu,\nu}$ when $\nu = p^r$?
• RQ5 Can the constants $\alpha_{\mu,\nu}$, counting solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $\sum z_i = 0$, be expressed via multinomial coefficients when $\nu$ is a prime power?

Key findings

• The maximum likelihood degree of $F_{n,d}$ is given by $\operatorname{MLdeg}(F_{n,d}) = \sum_{k=1}^n d^k - \sum_{j=0}^{n-1} \binom{n+1}{j} \beta_{n-j,d-1}$, where $\beta_{\mu,\nu}$ counts complex solutions to $z_1^\nu = \cdots = z_\mu^\nu = 1$ and $z_1 + \cdots + z_\mu + 1 = 0$.
• For $d=2$, the MLD is $\operatorname{MLdeg}(F_{n,2}) = 2^{n+1} - 2$, recovering a known result.
• For $n=2$, the MLD of $F_{2,d}$ depends on $d \mod 6$: it is $d^2 + d$ if $d \equiv 0,2 \pmod{6}$, $d^2 + d - 3$ if $d \equiv 3,5 \pmod{6}$, $d^2 + d - 2$ if $d \equiv 4 \pmod{6}$, and $d^2 + d - 5$ if $d \equiv 1 \pmod{6}$.
• When $d-1 = p^r$ is a prime power, the MLD has a closed formula involving multinomial coefficients: $\operatorname{MLdeg}(F_{n,d}) = \sum_{k=1}^n d^k - \frac{1}{d-1} \sum \frac{(n+1)!}{(n+1 - p(s_1+\cdots+s_k))! \cdot \prod_{i=1}^k (s_i!)^p}$, where the sum is over non-negative integers $s_1,\dots,s_k$ with $\sum s_i \leq \frac{n+1}{p}$ and $k = \frac{d-1}{p}$.
• The constants $\beta_{\mu,\nu}$ vanish when $\mu+1$ is not divisible by $p$ if $\nu = p^r$, and are given by a multinomial sum when divisible.
• The non-vanishing of $\alpha_{\mu,\nu}$ (and thus $\beta_{\mu,\nu}$) is characterized by Lam and Leung: $\alpha_{\mu,\nu} \neq 0$ iff $\mu$ lies in the semigroup generated by the prime divisors of $\nu$.