Quasi-Homogeneous Singularity

Updated 17 November 2025

Quasi-homogeneous singularity is defined by equations that admit a weighted grading, ensuring uniform scaling and canonical normal forms.
It is characterized by criteria such as Milnor-Tjurina equality and syzygetic conditions, which aid in classification across algebraic, analytic, and noncommutative frameworks.
These singularities are crucial in moduli problems and deformation theory, providing computational tools for integrable systems and symplectic geometry.

A quasi-homogeneous singularity is a singularity for which the defining analytic (or algebraic) equation admits a weighted grading under which it is homogeneous. This framework encompasses a wide spectrum of singularities in algebraic and analytic geometry, as well as in noncommutative algebra, symplectic geometry, and integrable systems. Quasi-homogeneous singularities are distinguished by their deep algebraic structure, their symmetry properties under scaling, and the existence of canonical normal forms, with implications for classification, moduli, and deformation theory across disciplines.

1. Algebraic and Analytic Definitions

A germ of an analytic function $f \in k\{x_1,\dots,x_n\}$ at the origin is called quasi-homogeneous if there exists a weight vector $(r_1, ..., r_n) \in \mathbb Q_{>0}^n$ and degree $d>0$ such that each monomial $x_1^{i_1} \cdots x_n^{i_n}$ appearing in $f$ with nonzero coefficient satisfies

$\sum_{j=1}^n r_j i_j = d.$

Equivalently, the scaling

$(x_1, ..., x_n) \mapsto (\lambda^{r_1} x_1, ..., \lambda^{r_n} x_n)$

acts on $f$ as $f(\lambda^{r_1} x_1, ..., \lambda^{r_n} x_n) = \lambda^d f(x_1,\dots,x_n)$ . In this case, the hypersurface singularity $V(f) = \{ f = 0 \}$ at the origin is a quasi-homogeneous singularity. The definition extends to parameterized germs, map-germs, and to the setting of noncommutative superpotentials by considering appropriate analogs of weights, gradings, and cyclic derivatives (Hassanzadeh et al., 21 Sep 2025, Hua et al., 2018).

2. Classical Characterization: Saito’s Theorem and Milnor-Tjurina Equality

For a hypersurface singularity $f$ , K. Saito’s result gives a precise criterion: $f$ is (right-equivalent to) a quasi-homogeneous germ if and only if it is contained in its own Jacobian ideal $J_f = (\partial f/\partial x_1, ..., \partial f/\partial x_n )$ , i.e., $f \in J_f$ (Hassanzadeh et al., 21 Sep 2025). Analytically, for an isolated singularity, this is further equivalent to the equality of two key invariants:

Milnor number $\mu_f = \dim_k k\{x\}/J_f$
Tjurina number $\tau_f = \dim_k k\{x\}/(J_f, f)$

A singularity is quasi-homogeneous if and only if $\mu_f = \tau_f$ . This holds for both analytic germs and formal power series, and generalizes to the commutative and noncommutative settings. In the noncommutative context (superpotentials), the corresponding Hochschild class must vanish (Hua et al., 2018).

3. Syzygetic, Foliation, and Computational Criteria

Modern approaches characterize quasi-homogeneity via modules of syzygies and logarithmic vector fields:

In both the local and graded/projective contexts, quasi-homogeneity can be detected by the existence of a syzygy of the Jacobian (or Tjurina) ideal whose evaluation at the singular point does not vanish (Andrade et al., 10 Feb 2025, Andrade et al., 8 Jul 2024, Hassanzadeh et al., 21 Sep 2025). Specifically, for a projective hypersurface $V=V(f)\subset\mathbb{P}^n$ , an isolated singular point $p$ is quasi-homogeneous if and only if there is a global Jacobian syzygy whose coordinate does not vanish at $p$ .
The module of essential logarithmic derivations, $\mathcal{E}_f$ , is cyclic if and only if all singular points are quasi-homogeneous (Hassanzadeh et al., 21 Sep 2025).
These conditions are efficiently checkable using computer algebra systems by evaluating the rank/dimension of syzygy matrices at singular points.

Furthermore, for plane curves that are free or nearly free, the vanishing of the first syzygy matrix at a point signals failure of quasi-homogeneity at that singularity (Andrade et al., 8 Jul 2024).

Table: Syzygy matrix rank criterion for quasi-homogeneity

Context	Matrix evaluated at $p$	Quasi-homogeneous iff
Projective hypersurface $V(f)$	First syzygy matrix $M_f(p)$	rank $(M_f(p)) \geq 1$
Plane free/nearly free curve	First syzygy matrix $M_f(p)$	rank $(M_f(p)) \geq 1$
Local analytic germ $f(x)$	Syzygy matrix $Z(0)$	Some entry is a unit

4. Structural and Classification Aspects

Quasi-homogeneous singularities exhibit structural properties reflected in their combinatorics and classification:

The weight system $(w_1, ..., w_n; d)$ and type encoded by a choice map $\kappa: \{1,\dots, n\} \to \{1,\dots, n\}$ (graph structure) dictate both the algebraic form and the geometry of the singularity. Any quasi-homogeneous singularity with isolated critical point can be written as $f = f_\kappa + f_{add}$ : a canonical part plus an admissible sum of monomials determined by combinatorial constraints (failing sets) (Hertling et al., 2010, Rarovskii, 27 May 2024).
Classification of types (Fermat, chain, cycle, exceptional) and graph combinatorics enables enumeration and systematic paper; for example, primes as Milnor numbers occur only in chain types (Hertling et al., 2010).
For two-dimensional cases, the analytic moduli are governed by the data $(p, q, n)$ and the configuration of parameters in a canonical factorization, modulo linear group actions (Câmara et al., 2014).

5. Symplectic and Noncommutative Quasi-Homogeneity

In the symplectic category, quasi-homogeneous parameterized singularities are studied via algebraic restrictions of closed $2$-forms. The new discrete invariant—the “proportional minimum quasi-degree part”—complements classical invariants for orbit classification and is stable under symplectic equivalence (Lira et al., 2016). The method of algebraic restrictions and local symplectic algebra provide complete invariants for quasi-homogeneous curves, with extensions to higher codimension.

In the noncommutative (superpotential) setting, a superpotential $W$ in a free algebra is quasi-homogeneous if and only if its class in the 0-th Hochschild homology of the Jacobi algebra vanishes. The corresponding noncommutative Saito theorem states that Jacobi-finiteness plus this vanishing is equivalent to being right-equivalent to a weighted homogeneous normal form. The weight vector is a formal invariant of right-equivalence (Hua et al., 2018).

6. Deformation Theory, Integrable Systems, and the Painlevé Property

The role of quasi-homogeneity extends to deformation and integrable system theory:

For quasi-homogeneous vector fields, the principal Laurent expansion generates a parameter space modeled as a Frobenius manifold, with the parameter weights governed by the Kovalevskaya exponents (Zhou et al., 30 May 2025).
Resonance conditions between Kovalevskaya exponents (often arithmetic in nature) signal the appearance of fractional powers or logarithmic terms in solutions, and the breakdown of strong forms of the Painlevé property.
In Hamiltonian models, pulled-back symplectic structures on the free parameter space force paired spectra of exponents, linking analytic, algebraic, and geometric properties across deformation classes.
The criterion of free parameter existence in Laurent expansions for such systems is a necessary and sufficient test for the integrability and absence of movable branch points.

7. Applications, Moduli, and Broader Context

Quasi-homogeneous singularities serve as building blocks for moduli problems and classification in singularity theory, algebraic geometry, and mathematical physics.

In moduli spaces, the set of quasi-homogeneous singularities with prescribed invariants (weights, Milnor number, syzygy type) forms a well-understood, sometimes complete, analytic or geometric classification space (Câmara et al., 2014).
In projective and combinatorial settings, the splitting of the Poincaré polynomial for free arrangements of plane curves with only quasi-homogeneous singularities yields Terao-type factorization results and combinatorial constraints on admissible intersection patterns (Pokora, 11 Dec 2024).
The concept generalizes to higher codimension, non-isolated singularities, parameterized surfaces (where it controls invariants like multiplicity and equisingularity), and noncommutative algebra.
In Landau-Ginzburg orbifold theory, equivalences between non-invertible quasi-homogeneous singularities and certain orbifold data are established via explicit isomorphisms of Frobenius algebras and matrix factorization categories (Rarovskii, 27 May 2024).

Quasi-homogeneous singularities thus act as central, canonical models throughout singularity and deformation theory, providing powerful algebraic, geometric, and computational tools for the analysis and classification of singular objects in diverse mathematical frameworks.