# Graduate/Postdoc Topology and Singularities Seminar

## Fall 2021 / Spring 2022

This year the seminar will continue to be on Zoom, hosted by Laurentiu Maxim and Botong Wang. The meeting information is below. We meet on Mondays, at 10am. Seminar announcements will be sent to the "singularities" mailing list. To subscribe, please send an email at: join-singularities@lists.wisc.edu (or, if you get an error message, just email maxim@math.wisc.edu). We plan to have all talks recorded.

The seminar is targeted at junior mathematicians with an interest in the topological study of singularities. We ask the speakers to take this fact into consideration when preparing their talks.

Topic: Topology and Singularities Seminar

Meeting ID: 997 5395 0856 Passcode: singular

date speaker title
Sept 10-11

TIBAR60 Conference

Sept 20 Botong Wang (UW-Madison) Perverse sheaves on varieties with large fundamental group and the Singer-Hopf conjecture
Sept 27 Rodolfo Aguilar Aguilar (Sofia, Bulgaria) Arrangements, fundamental groups and homology planes
October 4 Yongqiang Liu (USTC, China) $L^2$-type invariants and cohomology jump loci for smooth complex quasi-projective varieties
Dec 6 Jacob Matherne (Bonn) TBA

## Abstracts

### Botong Wang

The Singer-Hopf conjecture predicts that for a compact aspherical manifold $X$ of real dimension 2$n$, the Euler characteristic of $X$ has sign $(-1)^n$. In a joint work with Yongqiang Liu and Laurentiu Maxim, we made a stronger conjecture in the complex algebraic setting that if a smooth complex projective variety is aspherical, then all perverse sheaves on the variety has nonnegative Euler characteristics. We confirm this conjecture in a special case, when the fundamental group admits a faithful, semisimple, cohomologically rigid representation. Our approach is to construct a complex variation of Hodge structure on the variety and relate the nonnegativity of Euler characteristics with certain curvature conditions on the period domain. Joint work with Donu Arapura.

### Rodolfo Aguilar Aguilar

We will show some examples of homology planes: smooth, affine complex surfaces with trivial reduced integral homology, of log-general type which have an infinite fundamental group. These could be the first examples where such infinitude is shown, they arise as partial compactifications of the complement of an arrangement of lines in the complex projective plane.

### Irma Pallarés Torres

The Brasselet-Schürmann-Yokura conjecture for rational homology manifolds

Abstract: The Brasselet-Schürmann-Yokura conjecture is a conjecture of characteristic classes of singular spaces formulated by J. P. Brasselet, J. Schürmann, and S. Yokura. The conjecture predicts the equality between the Hodge L-class and the Goresky-MacPherson L-class for compact complex algebraic varieties that are rational homology manifolds. In this talk, I will explain two proofs of the conjecture. The first, via classical Hodge theory for projective varieties. This is a joint work with J. Fernández de Bobadilla. The second, using the theory of mixed Hodge modules for general compact algebraic varieties. This is a joint work with J. Fernández de Bobadilla and M. Saito.

### Manuel González Villa

On a quadratic form associated with the nilpotent part of the monodromy of a curve

Abstract: Joint work with Lilia Alanís-López, Enrique Artal Bartolo, Christian Bonatti, Xavier Gómez-Mont, and Pablo Portilla Cuadrado. We study the nilpotent part N of certain pseudo-periodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form Q defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form Q obtained after killing ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form Q is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to Q are able to distinguish plane curve singularities with different topological types but the same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space.

### Feng Hao

Holomorphic 1-forms, smoothness of morphisms to abelian varieties and some linearity properties

Abstract: In this talk I will discuss some results on how the existence of nowhere vanishing holomorphic 1-forms on a smooth complex projective variety X affect the singularities of morphisms from X to (simple) abelian varieties. Also, I will discuss some results on the linearity of the set of holomorphic 1-forms admitting zeros, which is related to the study of singularities of Albanese map. This is a joint work with Yajnaseni Dutta and Yongqiang Liu.

### Alex Suciu

Ab-exact extensions and Milnor fibrations of arrangements

Abstract: We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement.

### Rares Rasdeaconu

Moduli spaces of stable, rank 1, torsion free sheaves on real curves

Abstract: The counting of rational curves representing primitive homology classes on complex or real K3 surfaces is governed by the Yau-Zaslow formula and its real analog, respectively. A natural approach to extend such formulae to the non-primitive case requires the computation of the Euler characteristic of moduli spaces of stable, rank one sheaves on curves which are possibly reducible and non-reduced. The recent developments in this direction will be presented (joint work with V. Kharlamov).

Positive factorizations of monodromies on links of isolated complex surface singularities

Abstract: We present a generalization of a classical result concerning smooth germs of surfaces, by showing that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs are a product of right-handed Dehn twists. We explore some consequences of this theorem in the realm of mapping class groups of surfaces. If time permits, we will talk about the higher-dimensional counterpart of the result and how it possibly provides an obstruction to the smoothability of isolated singularities.

Minimal Exponents and a conjecture of Teissier

Abstract: The minimal exponent, defined by M. Saito, is an invariant of hypersurface singularities which is a refinement of the log canonical threshold. One expects singularities to worsen when intersecting with a smooth hypersurface, and Teissier conjectured in the 80's an inequality describing how much the minimal exponent can decrease in this process. I will describe joint work with Mircea Mustaţă (building on work he has done with Eva Elduque) in which we prove this conjecture.

### Daniel Bath

Towards a Logarithmic Comparison Theorem for Quasi-Free Divisors

Abstract: One way to compute the cohomology of the complement of a hypersurface {f=0} is to compute the cohomology of the de Rham complex of meromorphic forms with poles of arbitrary order along f. Sitting inside this complex is the logarithmic de Rham complex, consisting of certain forms with poles of order at most one. It is a long standing question to find necessary and sufficient conditions on the hypersurface ensuring that the natural inclusion of complexes is a quasi-isomorphism, that is, that f satisfies the Logarithmic Comparison Theorem. The modern approach utilizes D-module techniques and the "best" sufficient conditions require assuming that (among other things) the singular locus of f is, morally, as large as possible. To relax this assumption, we introduce a new variant of the Logarithmic Comparison Theorem where: logarithmic derivations are replaced with a submodule closed under Lie brackets; the logarithmic de Rham complex is replaced with a new de Rham complex. I will discuss our proposed strategy for proving this Quasi-Free Logarithmic Comparison Theorem which employs curious Bernstein--Sato polynomial-type constructions. Work in progress; joint with Luis Narvaez-Macarro and Francisco Castro-Jimenez.

### Dominik Wrazidlo

Intersection spaces and Poincaré completion

Abstract: In this survey talk, I will present a two-step method that assigns rational Poincaré duality spaces to certain compact oriented pseudomanifolds by modifying only a neighborhood of the singular set. The first step of the method consists of the intersection space construction, which was invented by Banagl, and was later extended by Agustín and Fernández de Bobadilla to a larger class of pseudomanifolds of arbitrary stratification depth. I will explain work in progress about the study of the rational homotopy type of intersection spaces. The second step of the method is a joint work with T. Essig, in which we extend intersection spaces to rational Poincaré duality spaces in certain cases by constructing a fundamental class. We achieve such a "Poincaré completion" by attaching a finite number of cells to the intersection space, which generalizes work of Klimczak. The focus of this talk lies on a survey of main ideas, examples, and possible future directions of research.

### Jörg Schürmann

(Equivariant) characteristic classes of singular toric varieties

Abstract: In this survey talk, we discuss for singular toric varieties different (equivariant) characteristic classes like Chern, Todd, L and Hirzebruch classes, with applications to (weighted) lattice points counting and Euler-MacLaurin type formulae for lattice polytopes. This is about joint work with Laurentiu Maxim.

### Jose Ignacio Cogolludo

Homological invariants of even Artin kernels

Abstract: Even Artin groups form a special family generalizing right-angle Artin groups that are associated with even-labeled graphs. Some of their properties as groups can be described in terms of their defining graph. In this talk we will discuss different properties associated with the homology of cocyclic subgroups and other finiteness conditions. This is a joint work in progress with C. Martínez-Pérez and R. Blasco-García.

Alexander modules and Mellin transform

Abstract: I will talk about the study of Alexander modules of algebraic varieties using Gabber and Loeser's Mellin transform. The main strength of this approach is that it allows the application of the full machinery of the theory of perverse sheaves, and even mixed Hodge modules. We obtain new results about the structure of Alexander modules, especially about their torsion part and, in the multivariable case, their artinian submodules. It also yields a mixed Hodge structure on the maximal artinian submodules of the Alexander modules. This is based on joint work with Eva Elduque, Christian Geske, Laurentiu Maxim and Botong Wang.

### Owen Barrett

The derived category of the abelian category of constructible sheaves

Abstract: Nori proved in 2002 that given a complex algebraic variety $$X$$, the bounded derived category of the abelian category of constructible sheaves on $$X$$ is equivalent to the usual triangulated category $$D(X)$$ of bounded constructible complexes on $$X$$. He moreover showed that given any constructible sheaf $$\mathcal F$$ on $$\mathbb{A}^n$$, there is an injection $$\mathcal F\hookrightarrow\mathcal G$$ with $$\mathcal G$$ constructible and $$H^i(\mathbb{A}^n,\mathcal G)=0$$ for $$i>0$$.

In this talk, I’ll discuss how to extend Nori’s theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $$\ell$$-adic sheaves. This is the case $$p=0$$ of the general problem which asks whether the bounded derived category of $$p$$-perverse sheaves is equivalent to $$D(X)$$, resolved affirmatively for the middle perversity by Beilinson.

### Ruijie Yang

Decomposition theorem for semisimple local systems

Abstract: In this talk, I would like to present a new and geometric proof of Sabbah's Decomposition Theorem, which asserts that any semisimple local system remains semisimple after taking the direct image under proper algebraic maps.

Sabbah's proof is D-module-theoretic in nature and relies on his theory of polarizable twistor D-modules, which generalizes Saito's theory of polarizable Hodge modules. Instead, we combine the topological approach developed by De Cataldo-Migliorini and Simpson's theory of mixed twistor structures to give the new proof. Along the way, we obtain new results about the cohomology of semisimple local systems. Joint work with Chuanhao Wei.

## Fall 2018

The Seminar meets at 10.30 to 11:30 on Fridays in Van Vleck 901.

date speaker title
Oct 5 Fenglin Li "Hasse principle and u-invariant"
Oct 26 Fenglin Li "Hasse principle and u-invariant (II)"
Nov 2 José Rodríguez "Maximum likelihood degree"

### Abstracts

#### Nov 2: José Rodríguez

Maximum likelihood degree

In statistics, point estimation uses sample data to calculate the "best estimate" of an unknown population parameter. For example, the sample average can be used to estimate the population mean. While there are many different point estimators, some of the most common ones are the maximum likelihood estimator (MLE), method of moments, and generalized method of moments (GMM).

In algebraic statistics statistical models are studied through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. In this talk, I will review maximum likelihood estimation and the maximum likelihood degree (ML degree) for discrete models. In particular, I will discuss how the ML degree gives a measure of algebraic complexity of the point estimate for MLE and how we can compute it using tools from topology and geometry. If time permits I will also discuss how we can use maximum likelihood degrees to study singularities.

## Fall 2017

The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.

date speaker title
Oct 4 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"
Oct 11 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"
Oct 18 Sebastian Baader "Dehn twist length in mapping class groups"
Oct 25 Cancelled
Nov 1 Christian Geske "Algebraic Intersection Spaces (I)"
Nov 8 Christian Geske "Algebraic Intersection Spaces (II)"
Nov 15 Laurentiu Maxim "Stratified Morse Theory: an overview (I)"
Nov 22 Thanksgiving break
Nov 29 Laurentiu Maxim "Stratified Morse Theory: an overview (II)"
December 6 Alexandra Kjuchukova "Singular branched covers of four-manifolds and applications"
December 13 TBD "TBA"

## Spring 2017

Fridays at 11:00 VV901

The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Jan 27 Christian Geske "Intersection Spaces and Equivariant Moore Approximation I"
Feb 3 Christian Geske "Intersection Spaces and Equivariant Moore Approximation II"
Feb 10 Sashka "The Wirtinger Number of a knot equals its bridge number I"
Feb 17 Sashka "The Wirtinger Number of a knot equals its bridge number II"
Feb 24 Christian Geske "Intersection Spaces and Equivariant Moore Approximation III"
Mar 3 Manuel Gonzalez Villa "Multiplier ideals of irreducible plane curve singularities"

## Fall 2016

Wednesdays at 14:30 VV901

The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Sept. 14 (W) Laurentiu Maxim "Alexander-type invariants of hypersurface complements"
Sept. 21 (W) Botong Wang "Cohomology jump loci"
Sept. 28 (W) Alexandra Kjuchukova "On the Bridge Number vs Meridional Rank Conjecture"
Oct 5 (W) Manuel Gonzalez Villa "Introduction to Newton polyhedra"
Oct 12 (W) Manuel Gonzalez Villa "More on Newton polyhedra"
Oct 26 (W) Christian Geske "Intersection Spaces"
Nov 2 (W) Christian Geske "Intersection Spaces Continued"
Nov 9 (W) CANCELLED
Nov 16 (W) Eva Elduque "Braids and the fundamental group of plane curve complements"
Nov 30 (W) Laurentiu Maxim "Novikov homology of hypersurface complements"
Dec 7 (W) CANCELLED
Dec 14 (W) Eva Elduque Specialty Exam: "Twisted Alexander invariants of plane curve complements"

## Spring 2016

Mondays at 3:20 B139VV

The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.

The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.

date speaker title
Feb. 8 (M) Christian Geske Section 1.1 and 1.2: Category of complexes and Homotopical category
Feb. 15 (M) Eva Elduque Sections 1.3 and 1.4: Derived category and derived functors
Feb. 22 (M) Botong Wang Sections 2.1 and 2.2: Generalities on Sheaves and Derived tensor products
Feb. 29 (M) Christian Geske Hypercohomology and Holomorphic Differential Forms on Analytic Varieties
Mar. 7 (M) Eva Elduque Section 2.3: Direct and inverse image
Mar. 14 (M) Cancelled
Mar. 28 (M) Cancelled
Apr. 4 (M) Cancelled
Apr. 11 (M) Christian Geske Section 2.3 cont.
Apr. 18 (M) Cancelled
Apr. 25 (M) Cancelled
May. 2 (M) Cancelled

## Abstracts

(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).

This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).

The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.

Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.

## Fall 2015

Thursdays 4pm in B139VV

date speaker title
Sept. 24 (Th) KaiHo (Tommy) Wong Twisted Alexander Invariant for Knots and Plane Curves
Oct. 1 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers I
Oct. 8 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers II
Oct. 15 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture I
Oct. 22 (Th) Yun Su (Suky) Pretalk Higher-order degrees of hypersurface complements., Survey on Alexander polynomial for plane curves.
Oct. 29 (Th) Yun Su (Suky) Aftertalk Higher-order degrees of hypersurface complements.
Nov. 5 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture II
Nov. 12 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture III
Nov. 19 (Th) Eva Elduque Stiefel-Whitney classes
Dec. 3 (Th) Eva Elduque Grass-mania!
Dec. 10 (Th) KaiHo (Tommy) Wong Pretalk Milnor Fiber of Complex Hyperplane Arrangements

## Abstracts

### Th, Sep 24: Tommy

Twisted Alexander Invariant of Knots and Plane Curves.

I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.

### Th, Oct 1 and 8: Sashka

Linking numbers and branched coverings I and II

Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves.

In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.

### Th, Oct 15, Nov 5 and Nov 12: Manuel

On poles of zeta functions and monodromy conjecture I and II

Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.

### Th, Nov 19: Eva

Stiefel-Whitney classes

Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.

### Th, Dec 3: Eva

Grass-mania!

In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.

This talk is for the most part self contained, so it doesn't matter if you missed the previous one.

### Th, Dec 10: Tommy

A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!

I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.

## Spring 2014

We continue with Professor Alex Suciu's work.

## Fall 2014

We follow Professor Alex Suciu's work this semester.

But we will not meet at a regular basis.

## Spring 2014

We meet on Tuesdays 3:30-4:25pm in room B211.

date speaker title
Feb. 25 (Tue) Yongqiang Liu Monodromy Decomposition I
Mar. 4 (Tue) Yongqiang Liu Monodromy Decomposition II
Mar. 25 (Tue) KaiHo Wong Conjecture of lower bounds of Alexander polynomial
Apr. 8 (Tue) Yongqiang Liu Nearby Cycles and Alexander Modules

## Fall 2013

We are learning Hodge Theory this semester and will be following three books:

1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II

2. Peters, Steenbrink, Mixed Hodge Structures

We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.

date speaker title
Sep. 18 (Wed) KaiHo Wong Discussions on book material
Sep. 25 (Wed) Yongqiang Liu Milnor Fibration at infinity of polynomial map
Oct. 9 (Wed) KaiHo Wong Discussions on book material
Oct. 16 (Wed) Yongqiang Liu Polynomial singularities
Nov. 13 (Wed) KaiHo Wong Discussions on book material

## Spring 2013

date speaker title
Feb. 6 (Wed) Jeff Poskin Toric Varieties III
Feb.13 (Wed) Yongqiang Liu Intersection Alexander Module
Feb.20 (Wed) Yun Su (Suky) How do singularities change shape and view of objects?
Feb.27 (Wed) KaiHo Wong Fundamental groups of plane curves complements
Mar.20 (Wed) Jörg Schürmann (University of Münster, Germany) Characteristic classes of singular toric varieties
Apr. 3 (Wed) KaiHo Wong Fundamental groups of plane curves complements II
Apr.10 (Wed) Yongqiang Liu Milnor fiber of local function germ
Apr.17 (Wed) 2:45pm-3:45pm (Note the different time) KaiHo Wong Formula of Alexander polynomials of plane curves

## Abstracts

### Wed, 2/27: Tommy

Fundamental groups of plane curves complements

I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed.

## Fall 2012

date speaker title
Sept. 18 (Tue) KaiHo Wong Organization and Milnor fibration and Milnor Fiber
Sept. 25 (Tue) KaiHo Wong Algebraic links and exotic spheres
Oct. 4 (Thu) Yun Su (Suky) Alexander polynomial of complex algebraic curve (Note the different day but same time and location)
Oct. 11 (Thu) Yongqiang Liu Sheaves and Hypercohomology
Oct. 18 (Thu) Jeff Poskin Toric Varieties II
Nov. 1 (Thu) Yongqiang Liu Mixed Hodge Structure
Nov. 15 (Thu) KaiHo Wong Euler characteristics of hypersurfaces with isolated singularities
Nov. 29 (Thu) Markus Banagl, University of Heidelberg High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres

## Abstracts

### Thu, 10/4: Suky

Alexander polynomial of complex algebraic curve

I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. From the definition, it is clear that Alexander polynomial is an topological invariant for curves. I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. Calculations of some examples will be provided.