[Paper Review] Multi-variable Poincaré series associated with Newton diagrams

Wolfgang Ebeling, S. M. Gusein‐Zade|ArXiv.org|May 30, 2009

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper introduces a multi-index filtration on the ring of germs of functions at a hypersurface singularity using its Newton diagram, generalizing quasi-homogeneous filtrations. It computes the multivariable Poincaré series for curve singularities and certain higher-dimensional cases, showing the series are of A’Campo type in these cases, though not always induced by a grading, as evidenced by negative coefficients in some examples.

ABSTRACT

We define a multi-index filtration on the ring of germs of functions on a hypersurface singularity associated with its Newton diagram and compute the multivariable Poincaré series of this filtration in some cases.

Motivation & Objective

• To define a multi-index filtration on the local ring of a hypersurface singularity based on its Newton diagram.
• To compute the multivariable Poincaré series of this filtration for specific classes of singularities.
• To investigate whether the Poincaré series are of A’Campo type or induced by a grading.
• To demonstrate that the Poincaré series are not always induced by a grading, using examples with negative coefficients.

Proposed method

• Define a multi-index filtration using order functions derived from the Newton diagram, where each variable corresponds to a monomial in the local ring.
• Compute the Poincaré series via the formula involving the dimension of quotient spaces $ d(\underline{v}) = \dim J(\underline{v})/J(\underline{v}+\underline{1}) $, and apply the standard Poincaré series formula involving $ \prod (t_i - 1) $ and $ t_1 \cdots t_s - 1 $.
• Use inclusion-exclusion and the Euler characteristic of projectivized quotient spaces to compute coefficients of the series.
• Analyze the structure of monomials in the Newton diagram and their images under the filtration, particularly focusing on points on the boundary of the Newton polyhedron.
• Apply a ${\mathbb{C}}^*$-action to compute the Euler characteristic of the projective space of non-vanishing components.
• Use the relation between the Poincaré series and the Newton diagram's facets and vertices to derive closed-form expressions, such as $ (1 - \underline{t}^{\underline{u}(f)}) \cdot P_{\{u_i\}}(\underline{t}) $.

Experimental results

Research questions

• RQ1 Can a multi-index filtration associated with a Newton diagram yield a Poincaré series of A’Campo type?
• RQ2 Is the Poincaré series of such a filtration always induced by a grading of the local ring?
• RQ3 What is the structure of the quotient space $ J(\underline{v})/J(\underline{v}+\underline{1}) $ in terms of monomials from the Newton diagram?
• RQ4 Can the coefficient of the Poincaré series be negative, indicating non-grading-induced structure?

Key findings

• For curve singularities and certain higher-dimensional singularities, the Poincaré series are of A’Campo type, expressed as $ (1 - \underline{t}^{\underline{u}(f)}) \cdot \prod_{i=1}^s (1 - \underline{t}^{\underline{u}(x_i)})^{-1} $.
• The coefficient at $ \underline{t}^{\underline{v}(f)} $ in the Poincaré series can be negative, as shown in the example with $ f(x,y,z) = x^5 + y^5 + z^5 + x^2yz + xy^2z + xyz^2 $, where the coefficient is $ -1 $.
• The Newton filtration in this example is not induced by a grading, as the Poincaré series coefficient is negative, contradicting the grading-induced case where coefficients are non-negative.
• In another example with $ f(x,y,z) = x^{20} + y^{20} + z^{16} + \cdots $, the coefficient at $ \underline{t}^{\underline{v}(f)} $ is $ 1 $, but the series is not of the standard A’Campo form due to the exponent vector not being a linear combination of the monomial valuations.
• The coefficient of the Poincaré series at $ \underline{t}^{\underline{v}} $ equals the Euler characteristic of the projectivized quotient space $ \mathbb{P}F_{\underline{v}} $, which can be computed via $ \mathbb{C}^* $-action fixed points.
• The structure of the quotient space $ J(\underline{v})/J(\underline{v}+\underline{1}) $ is freely generated by monomials corresponding to non-negative integer points on the boundary of the Newton polyhedron not dominated by the Newton diagram vertex.