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## WS 2015/2016

### 03.11.2015 um 16:15 Uhr in 69/125:

#### Prof. Dr. Amir Mafi (University of Kurdistan, Tehran, Iran)

##### Results on Ratliff-Rush ideal and reduction numbers

Let *(R, m)* be a Cohen-Macaualy local ring of positive dimension *d* and inﬁnite residue ﬁeld. Let *I* be an m primary ideal and J a minimal reduction of *I*. In this paper, we show that* r _{J}(I) ≤ r_{J }(I)*. This answer to a question that made by M.E. Rossi and I. Swanson in [1, Question 4.6].

### 10.11.2015 um 16:15 Uhr in 69/125:

#### Leo Margolis (Universität Stuttgart)

##### The Zassenhaus Conjecture for Integral Group Rings

Studying the unit group of the integral group ring * ZG* of a ﬁnite group *G*, there naturally arise questions about the structural connections of the unit group and the group base *G*. If, e.g., the units contain an element of some given order, does *G* contain an element of that order? Nowadays the main question concerning the ﬁnite subgroups of the unit group is the so called Zassenhaus Conjecture. It states that for any unit u in *ZG* of ﬁnite order there exist a unit x in the rational group algebra *QG* and an element *g* in * G*, such that

*x*or

^{−1}ux = g*x*. I will present the knowledge on this conjecture and related questions and some methods involved in attacking them. Particular attention will be paid to the so called HeLP-method, named after

^{−1}ux = −g**H**ertweck,

**L**uthar and

**P**assi, which may be regarded as an algorithm. This is a character theoretic method which for a given group

*G*ﬁnally reduces to a question on the solutions of integral linear inequalities.

### 24.11.2015 um 16:15 Uhr in 69/125:

#### Dr. Zafeirakis Zafeirakopoulos (Galatasaray University, Istanbul, Turkey)

##### Polyhedral Omega: A linear Diophantine system solver

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. In this way, we connect two branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. This is joint work with Felix Breuer.

### 08.12.2015 um 16:15 Uhr in 69/125:

**Prof. Dr. Hero Saremi** (Islamic Azad University, Sanandaj, Iran)

##### Hilbert regularity of Stanley-Reisner Rings

In this note, ﬁrstly, we characterize the Hilbert regularity of the Stanley-Reisner ring with respect to f-vector and h-vector of (d − 1)-dimensional simplicial complex ∆. As a consequence, we compute the Hilbert regular-ity of a boundary complex of a simplicial d-polytope.

### 15.12.2015 um 16:15 Uhr in 69/125:

#### Dr. Jan Uliczka (Universität Osnabrück)

##### Remarks on Hilbert series and Hilbert depth of graded modules

(with due regard to the *Z*^{2}–graded case)

^{2}

In this talk we consider finitely generated *Z ^{n}*-graded modules over a polynomial ring

*R=F[X*(over any field

_{1},...,X_{m}]*F*). The Hilbert series

*H*of such a module

_{M}(t) =∑_{a∈ Zn}(dim F M_{a})t_{a}*M*is known to be a rational function. We will investigate which formal Laurent series of this type occur as Hilbert seriesand provide some examples of such series which are

*not*the Hilbert series of afinitely generated module. Moreover, we will take a look at the notions of

*Hilbert depth*and

*Decomposition Hilbert depth*. The main result will be a criterion forpositive Hilbert depth of a

*Z*-graded module.

^{2}(Joint work with Lukas Katthän and Julio Moyano-Fernández)

### 12.01.2016 um 16:15 Uhr in 69/125:

#### Alessio Caminata (Universität Osnabrück)

##### Generalized Hilbert-Kunz function in graded dimension 2

Let *k* be a field and let *R* be a standard graded *k*-algebra. The generalized Hilbert-Kunz function and multiplicity have been introduced by Dao and Smirnov as an extension of the classical Hilbert-Kunz function and multiplicity to ideals which are not necessarily *R _{+}*-primary. In this talk I will focus on the two-dimensional situation and I will present a theorem which clarifies the structure of the generalized Hilbert-Kunz function, and extends a result proved by Brenner for the classical Hilbert-Kunz function. In particular, it follows that the generalized Hilbert-Kunz multiplicity is a rational number in this setting.

This is a joint work with Holger Brenner.

### 19.01.2016 um 16:15 Uhr in 69/125:

#### Dr. Jan Uliczka (Universität Osnabrück)

##### Hilbert depth of graded, namely ℤ^{2} graded, modules

This talk is a sequel to that of December 15^{th}. We will take a look at the notions of *Hilbert depth* and *Decomposition Hilbert dept*h for finitely generated ℤ^{n }gradedmodules over a polynomial ring R = F[X_{1},...,X_{m}]. The main result is a criterion for positive Hilbert depth of a ℤ^{2} graded module: the Hilbert depth is positive iff the coeffcients of the module's Hilbert series satisfy certain linear inequalities.

(Joint work with Lukas Katthän and Julio Moyano Fernández)